Morfismos, Vol 22, Num 2, 2018 by Morfismos, Department of Mathematics, Cinvestav

VOLUMEN 22 NÚMERO 2 JULIO A DICIEMBRE DE 2018 ISSN: 1870-6525

Morfismos, Volumen 22, Número 2, julio a diciembre 2018, es una publicación semestral editada por el Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav), a través del Departamento de Matemáticas. Av. Instituto Politécnico Nacional no. 2508, colonia San Pedro Zacatenco, Delegación Gustavo A. Madero, C.P. 07360, D.F., Tel. 55-57473800, www.cinvestav.mx, morfismos@math.cinvestav.mx, Editores Generales: Drs. Isidoro Gitler y Jesús González Espino Barros. Reserva de Derechos No. 04-2012-011011542900-102, ISSN: 1870-6525, ambos otorgados por el Instituto Nacional del Derecho de Autor. Certificado de Licitud de Tı́tulo No. 14729, Certificado de Licitud de Contenido No. 12302, ambos otorgados por la Comisión Calificadora de Publicaciones y Revistas Ilustradas de la Secretarı́a de Gobernación. Impreso por el Departamento de Matemáticas del Cinvestav, Avenida Instituto Politécnico Nacional 2508, Colonia San Pedro Zacatenco, C.P. 07360, México, D.F. Este número se terminó de imprimir en diciembre de 2018 con un tiraje de 50 ejemplares. Las opiniones expresadas por los autores no necesariamente reflejan la postura de los editores de la publicación. Queda estrictamente prohibida la reproducción total o parcial de los contenidos e imágenes de la publicación, sin previa autorización del Cinvestav.

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Proceedings for the 2nd International School on TQFT, Langlands and Mirror Symmetry 2013: Playa del Carmen

The Institute for Geometry and Physics Miami-Cinvestav-Campinas (IGP- MCC) is a joint virtual venture to stimulate a North-South collaboration in Geometry and Physics. The directors are Elisabeth Gasparim, Maxim Kontsevich, Ernesto Lupercio and Dennis Sullivan, and its founding coincided with the celebrations for the 50th anniversary of Cinvestav. Every year the institute organizes a conference in Miami, Florida to study the state of the art in Homological Mirror Symmetry. Every even numbered year Mexico organizes a school to bring top tier mathematicians to the country to study the state of the art in TQFT, Langlands and Mirror Symmetry. This volume represents the very best of the 2014 school.

Ernesto Lupercio1 Ludmil Katzarkov Denis Aroux Tony Pantev Invited Editors

1 E. L. Thanks the partial suport of the Moshinsky foundation and of a sabbatical at IMATE UNAM during the preparation of this volume.

Contents - Contenido On the equivariant De-Rham cohomology for non-compact Lie groups Camilo Arias Abad and Bernardo Uribe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin . . . . . . . . . . 7

Applications of gauged Gromov-Witten theory: a survey Eduardo GonzaĚ lez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Derived Mackey functors and profunctors: an overview of results D. Kaledin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Perverse Schobers Mikhail Kapranov and Vadim Schechtman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Sheaf of categories and categorical Donaldson theory Ludmil Katzarkov and Yijia Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Morfismos, Vol. 22, No. 2, 2018, pp. 1–6 Morfismos, Vol. 22, No. 2, 2018, pp. 1–6

On the equivarant De-Rham cohomology for non-compact Lie groups On the equivarant De-Rham cohomology for Lie groups Camilo non-compact Arias Abad Bernardo Uribe Camilo Arias Abad

Abstract

Bernardo Uribe

Let G be a connected and non-necessarily compact Lie group acting on the connected manifold M . In this short note we announce Abstract the Let following result: for a G-invariant closed compact differential onactG be a connected and non-necessarily Lieform group M , ing theon existence of a closed equivariant extension in the Cartan the connected manifold M . In this short note we announce model equivariant cohomology is equivalent the existence of on the for following result: for a G-invariant closedtodifferential form an extension in the homotopy quotient. M , the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of 2010 Mathematics Subject Classification: 57R91 an extension in the homotopy quotient. Keywords and phrases: Equivariant Cartan complex, non-compact Lie group, cohomology. 2010equivariant Mathematics Subject Classification: 57R91 Keywords and phrases: Equivariant Cartan complex, non-compact Lie group, equivariant cohomology.

Introduction

For1a manifold M endowed with the action of a Lie group G, the CarIntroduction tan model for the G-equivariant cohomology of the manifold M could be seen as athe De Rham forwith the equivariant cohomology. For manifold M version endowed the action of a Lie groupWhenever G, the Carthetan Liemodel group for G isthe compact, Cartan showed an equivariant version the be G-equivariant cohomology of the manifold M of could De seen Rham thusversion statingfor that G-equivariant cohomology of as Theorem, the De Rham thethe equivariant cohomology. Whenever thethe Cartan complex canonically isomorphic toequivariant the cohomology with Lie group G is is compact, Cartan showed an version of the realDecoefficients of the homotopy quotient ×G EG [1] cf.cohomology [4, Thm. of Rham Theorem, thus stating that theMG-equivariant 2.5.1]. Lie group is not compact, the G-equivariant cohomolthe When Cartanthe complex is canonically isomorphic to the cohomology with ogyreal of the Cartan complex fails to be isomorphic to the cohomology of coefficients of the homotopy quotient M ×G EG [1] cf. [4, Thm. the2.5.1]. homotopy quotient; therefore it was not possible to use topological When the Lie group is not compact, the G-equivariant cohomolproperties of the homotopy quotient to obtain information on of ogy of the Cartan complex fails toinbeorder isomorphic to the cohomology thethe Cartan differential forms. homotopy quotient; therefore it was not possible to use topological In this short notehomotopy we investigate the in relation between cohomol- on properties of the quotient order to obtainthe information ogythe of Cartan the G-equivariant Cartan complex of M and the cohomology differential forms. of the In homotopy ×G EG, the andrelation we announce this shortquotient note we M investigate betweenthat the indeed cohomology of the G-equivariant Cartan complex of M and the cohomology of the homotopy quotient M ×1G EG, and we announce that indeed 1

Camilo Arias Abad and Bernardo Uribe

there is a surjective map from the former to the latter. In particular this result implies that for a G-invariant closed differential form on M , the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient.

Equivariant Cartan complex for connected Lie groups

Let G be a connected Lie group with lie algebra g. Let K ⊂ G be the maximal compact subgroup of G and denote by k its Lie algebra. The inclusion of Lie algebras k → g induces a dual map g∗ → k∗ which is k-equivariant. Therefore we have the K-equivariant map S(g∗ ) → S(k∗ ) between the symmetric algebra on g∗ to the symmetric algebra on k∗ . Consider a manifold M endowed with an action of G. The Cartan complex associated to the G-manifold M is Ω∗G M := (S(g∗ ) ⊗ Ω∗ M )G , dG = d + Ωa ιXa where a runs over a base of g, Ωa denotes the element in g∗ dual to a and Xa is the vector field on M that defines the element a ∈ g. The composition of the natural maps (S(g∗ ) ⊗ Ω∗ M )G → (S(g∗ ) ⊗ Ω∗ M )K → (S(k∗ ) ⊗ Ω∗ M )K induces a homomorphism of Cartan complexes Ω∗G M → Ω∗K M. Theorem 2.1. Let G be a connected Lie group with Lie algebra g, let k be the Lie algebra of the maximal compact subgroup K of G and consider a G-manifold M . Then the map Ω∗G M → Ω∗K M induces a surjective map in cohomology H ∗ (Ω∗G M, dG ) H ∗ (Ω∗K M, dK ).

On the equivarant De-Rham cohomology for non-compact Lie groups

Since there are canonical isomorphisms H ∗ (Ω∗K M, dK ) ∼ = H ∗ (M ×K ∗ EK, R) ∼ = H (M ×G EG, R), we conclude that the canonical map H ∗ (Ω∗G M, dG ) H ∗ (M ×G EG, R) is surjective. Sketch of proof. Consider the complex C k (G, S(g∗ ) ⊗ Ω• M ) defined in [3, Section 2.1] whose elements are smooth maps f (g1 , . . . , gk |X) : Gk × g → Ω• M, which vanish if any of the arguments gi equals the identity of G. These maps could also be seen as maps f : Gk → S(g∗ ) ⊗ Ω• M, and whenever the map has for image a homogeneous polynomial of degree l, then its total degree is deg(f ) = k + l. The differentials d and ι are defined by the formulas (df )(g1 , . . . , gk |X) = (−1)k df (g1 , . . . , gk |X)

and

(ιf )(g1 , . . . , gk |X) = (−1) ι(X)f (g1 , . . . , gk |X), as in the case of the differentials in Cartan’s model for equivariant cohomology [1, 4]. The differential d¯ : C k → C k+1 is defined by the formula ¯ )(g0 , . . . , gk |X) = f (g1 , . . . , gk |X) (df +

k i=1

(−1)i f (g0 , . . . , gi−1 gi , . . . , gk |X)

+(−1)k+1 gk f (g0 , . . . , gk−1 |Ad(gk−1 )X), and the fourth differential ῑ : C k → C k−1 is defined by the formula (ῑf )(g1 , . . . , gk−1 |X) =

k−1 i=0

(−1)i

∂ f (g1 , . . . , gi , etXi , gi+1 . . . , gk−1 |X), ∂t

where Xi = Ad(gi+1 . . . gk−1 )X. The structural maps d, ι, d¯ and ῑ are all of degree 1, and the operator dG = d + ι + d¯ + ῑ

Camilo Arias Abad and Bernardo Uribe

becomes a degree 1 map that squares to zero. The cohomology of the complex (C ∗ (G, S(g∗ ) ⊗ Ω• M ), dG ) will be denoted by H ∗ (G, S(g∗ ) ⊗ Ω• M ) and in [3, Thm. 2.2.3] it was shown that there is a canonical isomorphism of rings H ∗ (G, S(g∗ ) ⊗ Ω• M ) ∼ = H ∗ (M ×G EG; R) Note that there are natural maps of complexes C ∗ (G, S(g∗ ) ⊗ Ω∗ M ) → C ∗ (K, S(k∗ ) ⊗ Ω∗ M ) inducing an isomorphism on its cohomology groups ∼ =

H ∗ (G, S(g∗ ) ⊗ Ω∗ M ) → H ∗ (K, S(k∗ ) ⊗ Ω∗ M ). This isomorphism follows from the fact that the inclusion K ⊂ G is a homotopy equivalence inducing a homotopy equivalence M ×K EK M ×G EG and the fact that H ∗ (M ×G EG, R) ∼ = H ∗ (G, S(g∗ ) ⊗ Ω∗ M ) for any connected Lie group G. Filtering the double complex C ∗ (G, S(g∗ ) ⊗ Ω∗ M ) by the degree of the elements in S(g∗ ) ⊗ Ω∗ M we obtain a spectral sequence whose first page is E1 = Hd∗ (G, S(g∗ ) ⊗ Ω∗ M ), the differentiable cohomology of G with values in the graded representation S(g∗ ) ⊗ Ω∗ M . Note that in the 0-th row we obtain E1∗,0 = (S(g∗ ) ⊗ Ω∗ M )G = Ω∗G M. The same degree filtration applied to the complex C ∗ (K, S(k∗ ) ⊗ produces a spectral sequence which at the first page is E 1 = ∗ Hd (K, S(k∗ ) ⊗ Ω∗ M ), and since K is compact this simply becomes Ω∗ M )

∗,0

E 1 = (S(k∗ ) ⊗ Ω∗ M )K = Ω∗K M

On the equivarant De-Rham cohomology for non-compact Lie groups

p,q

with E 1 = 0 for q = 0. The first differential of the spectral sequence once restricted to the 0-th row E1∗,0 = Ω∗G M is precisely the differential of the Cartan complex; therefore we obtain E2∗,0 = H ∗ (Ω∗G M ). Equivalently we obtain ∗,0

E 2 = H ∗ (Ω∗K M ) ∼ = H ∗ (M ×K EK, R), but in this case the spectral sequence collapses at the second page and ∗,0 the only non zero elements in E ∞ appear on the 0-th row E ∞ ∼ = H ∗ (M ×K EK, R). The canonical map between the complexes C ∗ (G, S(g∗ ) ⊗ Ω∗ M ) → C ∗ (K, S(k∗ ) ⊗ Ω∗ M ) induces a map of spectral sequences E• → E • , and we know that at ∗, = ∗, ∼ → E∞ . the pages at infinity it should induce an isomorphism E∞ Therefore the map ∗,0 E2∗,0 → E 2 must be a surjective map, and hence we have the canonical map ∗,0

Ω∗G M = E1∗,0 → E 1 = Ω∗K M inducing the desired surjective map in cohomology H ∗ (Ω∗G M, dG ) H ∗ (Ω∗K M, dK ). The complete and detailed proof of the previous theorem, as well as its applications, will appear in a forthcoming publication. Finally, from the previous theorem we may conclude: Corollary 2.2. Consider a G-invariant closed differential form on M . This differential from may be extended to a closed G-equivariant differential form in the Cartan complex of M if and only if the cohomology class of the differential form may be extended to a cohomology class in the homotopy quotient M ×G EG. This result generalizes similar statements that appeared in [2] in the case that the Lie group is reductive.

Camilo Arias Abad and Bernardo Uribe

Camilo Arias Abad Max-Planck-Institut fuer Mathematik, Vivatsgasse 7, 53111 Bonn, Germany camiloariasabad@gmail.com

Bernardo Uribe Departamento de Matemáticas y Estadı́stica, Universidad del Norte, Km. 5 vı́a Puerto Colombia, Barranquilla, Colombia buribe@gmail.com

References [1] Henri Cartan. La transgression dans un groupe de Lie et dans un espace fibré principal. In Colloque de topologie (espaces fibrés), Bruxelles, 1950, pages 57–71. Georges Thone, Liège; Masson et Cie., Paris, 1951. [2] Hugo Garcı́a-Compeán, Pablo Paniagua, and Bernardo Uribe. Equivariant extensions of differential forms for non-compact lie groups. In The influence of Solomon Lefschetz in Geomerty and Topology- 50 years of CINVESTAV, volume 621 of Contemp. Math., pages 19–33. Amer. Math. Soc., Providence, RI, 2014. [3] Ezra Getzler. The equivariant Chern character for non-compact Lie groups. Adv. Math., 109(1):88–107, 1994. [4] Victor W. Guillemin and Shlomo Sternberg. Supersymmetry and equivariant de Rham theory. Mathematics Past and Present. Springer-Verlag, Berlin, 1999. With an appendix containing two reprints by Henri Cartan [ MR0042426 (13,107e); MR0042427 (13,107f)].

Morfismos, Vol. 22, No. 2, 2018, pp. 7–20 Morfismos, Vol. 22, No. 2, 2018, pp. 7–20

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint Symplectic Lefschetz fibrations from a LieE.theoretical viewpoint B. Callander Gasparim L. Grama L. A. B. San Martin B. Callander E. Gasparim L. Grama L. A. B. San Martin

Abstract This is an announcement of results proved in [15], [16], [10], and [11] where methods from Lie theory were used as new tools for the Abstract study of symplectic Lefschetz fibrations. This is an announcement of results proved in [15], [16], [10], and [11] where methods from Lie theory were used as new 14D06. tools for the 2010 Mathematics Subject Classification: 22F30, 53D05, study of symplectic Lefschetz fibrations. Keywords and phrases: Adjoint orbit, Hodge diamonds, symplectic Lefchetz fibration. 2010 Mathematics Subject Classification: 22F30, 53D05, 14D06. Keywords and phrases: Adjoint orbit, Hodge diamonds, symplectic Leffibration. 1 chetz Introduction

A motivation for studying symplectic Lefschetz fibrations is that, in nice 1 Introduction cases, they occur as mirror partners of complex varieties. In fact, given a complex varietyforYstudying , the Homological Symmetry A motivation symplecticMirror Lefschetz fibrations(HMS) is that,conin nice jecture of Kontsevich [21] predicts the existence of a symplectic mirror cases, they occur as mirror partners of complex varieties. In fact, given partner X withvariety a superpotential W : X → Mirror C. ForSymmetry Fano varieties, thecona complex Y , the Homological (HMS) statement HMS includes the following: The category of A-branes jecture of of the Kontsevich [21] predicts the existence of a symplectic mirror D(Lag(W )) is equivalent to the derived category of B-branes (coherent partner X with a superpotential W : X → C. For Fano varieties, the sheaves) Db (Coh(X)) on X. Here the D(Lag(W )) is The the directed statement of the HMS includes following: category Fukaya– of A-branes Seidel category of vanishing cycles for the symplectic manifold and D(Lag(W )) is equivalent to the derived category of B-branesX(coherent b b is the bounded category of coherent Y. D (Coh(Y sheaves))) D (Coh(X)) on derived X. Here D(Lag(W )) is the sheaves directedon Fukaya– An Seidel exciting featureofofvanishing the conjecture A-side is symplectic category cycles is forthat the the symplectic manifold X and whereas the B-side is algebraic, and therefore theofconjecture provideson a Y. )) is the bounded derived category coherent sheaves Db (Coh(Y dictionary between the two types of geometry – algebraic andissymplecAn exciting feature of the conjecture is that the A-side symplectic tic –whereas the mirror map interchanging vanishing cycles on the symplectic the B-side is algebraic, and therefore the conjecture provides a sidedictionary with coherent sheaves on the algebraic side. – algebraic and symplecbetween the two types of geometry HMS has mirror been described in several cases: elliptic curves curves tic – the map interchanging vanishing cycles on [22], the symplectic of genus two [23], curves of higher genus [13], punctured spheres [2], side with coherent sheaves on the algebraic side. HMS has been described in several cases: elliptic curves [22], curves of genus two [23], curves of higher genus [13], punctured spheres [2], 7 7

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

weighted projective planes and del-Pezzo surfaces [6], [7], quadrics and intersection of two quadrics [27], the four torus [4], Calabi–Yau hypersurfaces in projective space [26], toric varieties [1], Abelian varieties [14], hypersurfaces in toric varieties [3], varieties of general type [17], and non-Fano toric varieties [9]. Nevertheless, the HMS conjecture remains open in most cases. The B-side of the conjecture is better understood in the sense that a lot is known about the category of coherent sheaves on algebraic varieties. In particular, in the Fano and general type cases, the famous reconstruction theorem of Bondal and Orlov says that you can recover the variety from its derived category of coherent sheaves [8]. In contrast the A-side is rather mysterious. The intent of this paper is to contribute to the understanding of LG models and subsequently to their categories of vanishing cycles. Using Lie theory, we construct LG models (O(H0 ), fH )), where O(H0 ) is the adjoint orbit of a complex semisimple Lie group and fH is the height function with respect to an element of the Cartan subalgebra (see Theorem 6.1). Even though we had HMS as an encouragement to pursue our work, we do not attempt to prove any instance of it, rather we endeavour to contribute to the understanding of the A-side of the conjecture by describing examples of symplectic Lefschetz fibrations in arbitrary dimensions. We calculate the directed Fukaya–Seidel category in the first nontrivial example, namely the adjoint orbit of sl(2, C). For the case of sl(3, C) orbits, we discuss (the wild) variations of Hodge diamonds depending on choices of compactifications for our Lefschetz fibrations.

Definitions

Definition 2.1. A holomorphic Morse function on a manifold X is a holomorphic function f : X → P1 (or f : X → C) which has only nondegenerate critical points. Definition 2.2. Let X be a complex manifold of dimension n and f : X → P1 (or f : X → C) a surjective holomorphic fibration. We say that f is a topological Lefschetz fibration if 1. there are finitely many critical points p1 , . . . , pk , and f (pi ) = f (pj ) for i = j; 2. for each critical point p, there are complex neighbourhoods p ∈ U ⊂ X, f (p) ∈ V ⊂ P1 on which f|U is represented by the holo-

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

morphic Morse function f|U (z1 , . . . , zn ) = z12 + . . . + zn2 , and such that crit f ∩ U = {p}; and 3. the restriction freg := f |X− Xi to the complement of the singular fibres Xi is a locally trivial fibre bundle. Definition 2.3. Let X be a complex manifold and ω a symplectic form making (X, ω) into a symplectic manifold. We say that a topological Lefschetz fibration is a symplectic Lefschetz fibration if 1. the smooth part of any fibre is a symplectic submanifold of (X, ω); and 2. for each critical point pi , the form ωpi is non-degenerate on the tangent cone of Xi at pi .

Non-examples

Proposition 3.1. Let M be a compact complex manifold with odd Euler characteristic, then M does not fibre over P1 . Proof. The Euler characteristic is multiplicative, that is, for such a fibration we would have χ(M ) =χ(P1 ) · χ(F ), but χ(P1 ) = 2. Corollary 3.2. [10, cor 2.19] For n > 1, there are no topological fibrations f : P2n → P1 . Proposition 3.3. There are no algebraic fibrations f : Pn → P1 for n > 1. Proof. Fibres of such a fibration would divisors be in Pn , but by Bezout theorem any two divisors in Pn intersect.

Good examples in dimension 4

In 4 (real) dimensions, every symplectic manifold admits a Lefschetz fibration after blowing up finitely many points. This is the celebrated result of Donaldson [12]: For any symplectic 4-manifold X, there exists a nonnegative integer n such that the n-fold blowup of X, topologically X#nCP2 , admits a Lefschetz fibration f : X#nCP2 → S 2 .

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

In the opposite direction, still in 4D, the existence of a topological Lefschetz fibration on a symplectic manifold guarantees the existence of a symplectic Lefschetz fibration whenever the fibres have genus at least 2 [19]: If a 4-manifold X admits a genus g Lefschetz fibration f : X → C with g ≥ 2, then it has a symplectic structure. Moreover, the existence of 4D symplectic Lefschetz fibrations with arbitrary fundamental group is guaranteed by [5]: Let Γ be a finitely presentable group with a given finite presentation a : πg → Γ. Then there exists a surjective homomorphism b : πh → πg for some h ≥ g and a symplectic Lefschetz fibration f : X → S 2 such that the regular fibre of f is of genus h, π1 (X) = Γ, and the natural surjection of the fundamental group of the fibre of f onto the fundamental group of X coincides with a ◦ b. In general it is possible to construct Lefschetz fibrations in 4D starting with a Lefschetz pencil and then blowing up its base locus (see [24], [25] [18]). However, in such cases one needs to fix the indefiniteness of the symplectic form over the exceptional locus by glueing in a correction. Direct constructions of Lefschetz fibrations in higher dimensions are by and large lacking in the literature. This gave us our first motivation to investigate the existence of symplectic Lefschetz fibrations on complex n-folds with n ≥ 3. Our construction does not make use of Lefschetz pencils, we construct our symplectic Lefschetz fibrations directly by taking the height functions that come naturally from the Lie theory viewpoint.

A caveat about the norm of complex Morse functions

It is sometimes claimed in the literature that |f |2 is a real Morse function whenever f is a Lefschetz fibration. However, this is in general false. We state this fact as a lemma. Lemma 5.1. Let X be a complex manifold of dimension, f : X → C a Lefschetz fibration and let p be a critical point of f . Then p is a degenerate critical point of |f − f (p)|2 . Proof. We may choose (complex) charts centred at p such that with respect to this coordinate system f (z1 , . . . , zn )−f (p) = ni=1 zi2 . Hence, is it enough to consider the standard Lefschetz fibration g : Cn → C given by g(z1 , . . . , zn ) = ni=1 zi2 , and to prove that 0 is a degenerate

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

critical point of |g|2 . In real coordinates n

z := (z1 , . . . , zn ) â&#x2020;&#x2019;

zi2

i=1

where we have written zi = xi +

n i=1

â&#x2C6;&#x161;

â&#x2C6;&#x161; x2i â&#x2C6;&#x2019; yi2 + 2 â&#x2C6;&#x2019;1xi yi ,

â&#x2C6;&#x2019;1yi . Then we have the function

|g|2 : Cn â&#x2020;&#x2019; R z â&#x2020;&#x2019;

whose differentials are 2

x2i

i=1

â&#x2C6;&#x201A;xk |g| = 4xk 2

â&#x2C6;&#x2019;

yi2

â&#x2C6;&#x201A;yk |g| = â&#x2C6;&#x2019;4yk

x2i

i=1

â&#x2C6;&#x2019;

x2i

i=1

x i yi

+ 8yk

i=1

yi2

â&#x2C6;&#x2019;

yi2

x i yi ,

i=1

+ 8xk

x i yi .

i=1

Since crit |g|2 â&#x160;&#x192; g â&#x2C6;&#x2019;1 (0), any neighbourhood of 0 contains a non-zero critical point of |g|2 and it follows that 0 is a degenerate critical point of |g|2 .

SLFs in higher dimensions via Lie theory

Let g be a complex semisimple Lie algebra with Cartan subalgebra h, and hR the real subspace generated by the roots of h. An element H â&#x2C6;&#x2C6; h is called regular if Îą (H) = 0 for all Îą â&#x2C6;&#x2C6; Î . Theorem 6.1. [15, thm. 3.1] Given H0 â&#x2C6;&#x2C6; h and H â&#x2C6;&#x2C6; hR with H a regular element, the potential fH : O (H0 ) â&#x2020;&#x2019; C defined by fH (x) = H, x

x â&#x2C6;&#x2C6; O (H0 )

has a finite number of isolated singularities and defines a Lefschetz fibration; that is to say 1. the singularities are (Hessian) nondegenerate; â&#x2C6;&#x2019;1 2. if c1 , c2 â&#x2C6;&#x2C6; C are regular values then the level manifolds fH (c1 ) â&#x2C6;&#x2019;1 and fH (c2 ) are diffeomorphic;

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

3. there exists a symplectic form Ω on O (H0 ) such that the regular fibres are symplectic submanifolds; 4. each critical fibre can be written as the disjoint union of affine subspaces contained in O (H0 ), each symplectic with respect to Ω. The full proof is presented in [15], a particularly interesting component of the proof states: Proposition 6.2. [15, prop. 3.3] A point x ∈ O(H0 ) is a critical point of fH if and only if x ∈ O (H0 ) ∩ h = W · H0 , where W is the Weyl group. Having found a construction of Lefschetz fibrations in higher dimensions, the next step toward a description of the Fukaya–Seidel category of the corresponding LG model would involve the identification of the Fukaya category of a regular fibre. Thus, we studied the diffeomorphism type of a regular level for the Lefschetz fibration. This first required the realisation of the adjoint orbit as the cotangent bundle of a flag manifold, as we now describe. We choose a set of positive roots Π+ and simple roots Σ ⊂ Π+ with corresponding Weyl chamber is a+ . A subset Θ ⊂ Σ defines a parabolic subalgebra pΘ with parabolic subgroup PΘ and a flag manifold FΘ = G/PΘ . An element HΘ ∈ cla+ is characteristic for Θ ⊂ Σ if Θ = {α ∈ Σ : α (HΘ ) = 0}. Let ZΘ = {g ∈ G : Ad (g) HΘ = HΘ } be the centraliser in G of the characteristic element HΘ . Theorem 6.3. [16, thm. 2.1] The adjoint orbit O (HΘ ) = Ad (G) · HΘ ≈ G/ZΘ of the characteristic element HΘ is a C ∞ vector bundle over FΘ isomorphic to the cotangent bundle T ∗ FΘ . Moreover, we can write down a diffeomorphism ι : Ad (G) · HΘ → T ∗ FΘ such that 1. ι is equivariant with respect to the actions of K, that is, for all k ∈ K, ι ◦ Ad (k) = k◦ι where K is the compact subgroup in the Iwasawa decomposition G = KAN , and k is the lifting to T ∗ FΘ (via the differential) of the action of k on FΘ .

2. The pullback of the canonical symplectic form on T ∗ FΘ by ι is the (real) Kirillov–Kostant–Souriaux form on the orbit.

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

Viewing the orbit as the cotangent bundle of a flag manifold, we can identify the topology of the of the fibres in terms of the topology of the flag. Corollary 6.4. [15, cor. 4.5] The homology of a regular fibre coincides with the homology of FΘ \W ·HΘ . In particular the middle Betti number is k − 1 where k is the number of singularities of the fibration (equal to the number of elements in W · HΘ ). For the case where singular fibres have only one critical point, we have the following corollary. Corollary 6.5. [15, cor. 5.1] The homology of the singular fibre though wHΘ , w ∈ W, coincides with that of FHΘ \ {uHΘ ∈ W cdotHΘ |u = w}. In particular, the middle Betti number of this singular fibre equals k − 2, where k is the number of singularities of the fibration fH .

Compactifications and their Hodge diamonds

Theorem 6.3 makes it clear that the adjoint orbits considered here are not compact. We want to compare the behaviour of vanishing cycles on O(H0 ) and on its compactifications. Expressing the adjoint orbit as an algebraic variety, we homogenise its ideal to obtain a projective variety, which serves as a compactification. We calculate the sheafcohomological dimensions dim H q (X, Ωp ) for the compactified orbits as well as for the fibres of the SLF. These dimensions shall be called the diamond for the given space; indeed, this is well-known as the Hodge diamond in the non-singular case. Calculating such diamonds is computationally heavy, so we used Macaulay2. Choosing a compactification is in general a delicate task: a different choice of generators for the defining ideal of the orbit can result in completely different diamonds of the corresponding compactification, as example 1 will show. To illustrate the behaviour of diamonds, we present some examples of adjoint orbits for sl(3, C), for which there are three isomorphism types. We chose one that compactifies smoothly and another whose compactification acquires degenerate singularities.

7.1

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

An SLF with 3 critical values

In sl(3, C), consider the orbit O(H0 ) of   2 0 0 H0 =  0 −1 0  0 0 −1 under the adjoint action. We fix the element   1 0 0 H =  0 −1 0  0 0 0

to define the potential fH . A general element A ∈ sl (3, C) has the form   x1 y 1 y2 . y3 (1) A =  z1 x 2 z2 z3 −x1 − x2

In this example, the adjoint orbit O(H0 ) consists of all the matrices with the minimal polynomial (A + id)(A − 2 id). So the orbit is the affine variety cut out by the ideal I generated by the polynomial entries of (A+id)(A−2 id). To obtain a projectivisation of X, we first homogenise its ideal I with respect to a new variable t, then take the corresponding projective variety. In this case, the projective variety X is a smooth compactification of X and has Hodge diamond: 1 0 0 0 0

0 2

0 0

3 0

0 0

0 2

0 . 0

0 0

1 We now calculate the Hodge diamond of a compactified regular fibre. The potential corresponding to our choice of H is fH = x1 − x2 . The critical values of this potential are ±3 and 0. Since all regular fibres of an SLF are isomorphic, it suffices to chose the regular value 1. We then define the regular fibre X1 as the variety in sl(3, C) ∼ = C8 corresponding

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

to the ideal J obtained by summing I with the ideal generated by fH â&#x2C6;&#x2019;1. We then homogenise J to obtain a projectivisation X 1 of the regular fibre X1 . The Hodge diamond of X 1 is: 1 0 0 0

0 2

0 0

2 0

0 . 0

0 1

Remark 7.1.1. An interesting feature to observe here is the absence of middle cohomology for the regular fibre. Suppose that the potential extended to this compactification without degenerate singularities, then because fH has singularities, the fundamental lemma of Picardâ&#x20AC;&#x201C; Lefschetz theory would imply that there must exist vanishing cycles, which contradicts the absence of middle homology. Generalising this example to the case of sl(n, C), we obtained: Proposition 7.1.2. [11, Prop. 2] Let H0 = Diag(n, â&#x2C6;&#x2019;1, . . . , â&#x2C6;&#x2019;1). Then the orbit of H0 in sl(n + 1, C) compactifies holomorphically to a trivial product. Corollary 7.1.3. [11, Cor 3] Choose H = Diag(1, â&#x2C6;&#x2019;1, 0, . . . , 0) and H0 = Diag(n, â&#x2C6;&#x2019;1, . . . , â&#x2C6;&#x2019;1) in sl(n+1, C). Any extension of the potential fH to the compactification Pn Ă&#x2014; Pnâ&#x2C6;&#x2014; of the orbit O(H0 ) cannot be of Morse type; that is, it must have degenerate singularities.

7.2

An SLF with 4 critical values

In sl (3, C) we take ďŁŤ

ďŁś 1 0 0 H = H0 = ďŁ 0 â&#x2C6;&#x2019;1 0 ďŁ¸ , 0 0 0

which is regular since it has 3 distinct eigenvalues. Then X = O (H0 ) is the set of matrices in sl (3, C) with eigenvalues 1, 0, â&#x2C6;&#x2019;1. This set forms a submanifold of real dimension 6 (a complex threefold). In this case W S3 acts via conjugation by permutation matrices. Therefore,

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

the potential fH = x1 â&#x2C6;&#x2019; x2 has 6 singularities; namely, the 6 diagonal matrices with diagonal entries 1, 0, â&#x2C6;&#x2019;1. The four singular values of fH are Âą1, Âą2. Thus, 0 is a regular value for fH . Let A â&#x2C6;&#x2C6; sl(3, C) be a general element written as in (1), and let p = det(A), q = det(A â&#x2C6;&#x2019; id). The ideals p, q and p â&#x2C6;&#x2019; q, q are clearly identical and either of them defines the orbit though H0 as an affine variety in sl (3, C). Now I = p, q, fH

J = p, p â&#x2C6;&#x2019; q, fH

are two identical ideals cutting out the regular fibre X0 over 0. Let Ihom and Jhom be the respective homogenisations and notice that Ihom = Jhom , so that they define distinct projective varieties, and thus two distinct compactifications I

X 0 = Proj(C[x1 , x2 , y1 , y2 , y3 , z1 , z2 , z3 , t]/Ihom ) J X0

and

= Proj(C[x1 , x2 , y1 , y2 , y3 , z1 , z2 , z3 , t]/Jhom )

of X0 . Their diamonds are given in Figure 1. 1

1 0 0 0 0 0

1 0

0 16

0 0

0 0 0

0 0

1 0

0 0

? 1

0 0

0 1

0 0

1 0

0 0

0 1

0 0

0 1 I

Figure 1: The diamonds of two projectivisations X 0 (left) and X 0 (right) of the regular fibre corresponding to H = H0 = Diag(1, â&#x2C6;&#x2019;1, 0). J

Remark 7.2.1. The variety X 0 is an irreducible component of X 0 . Indeed, we find that I â&#x160;&#x201A; J and that J is a prime ideal (whereas I is not). The discrepancy of values in the middle row is corroborated by the discrepancy between the expected Euler characteristics of the compactifications. Remark 7.2.2. Macaulay2 greatly facilitates cohomological calculations that are unfeasible by hand. However, the memory requirements

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

rise steeply with the dimension of the variety. The unknown entries in our diamonds (marked with a â&#x20AC;&#x2DC;?â&#x20AC;&#x2122;) exhausted the 48GB of RAM of the computers of our collaborators at IACS Kolkata, without producing an answer. Question. This leaves us with the open question of characterising all the compactifications of a given orbit produced by the method of homogenising the defining ideals. Nevertheless, once again methods of Lie theory provide us with sharper tools, and we obtain a compactification that is natural from the Lie theory viewpoint. Let w0 be the principal involution of the Weyl group W, that is, the element of highest length as a product of simple roots. For a subset Î&#x2DC; â&#x160;&#x201A; ÎŁ we put Î&#x2DC;â&#x2C6;&#x2014; = â&#x2C6;&#x2019;w0 Î&#x2DC; and refer to FÎ&#x2DC;â&#x2C6;&#x2014; as the flag manifold dual to FÎ&#x2DC; . If HÎ&#x2DC; is a characteristic element for Î&#x2DC; then â&#x2C6;&#x2019;w0 HÎ&#x2DC; is characteristic for Î&#x2DC;â&#x2C6;&#x2014; . Then the diagonal action of G on the product FÎ&#x2DC; Ă&#x2014; FÎ&#x2DC;â&#x2C6;&#x2014; as (g, (x, y)) â&#x2020;&#x2019; (gx, gy), g â&#x2C6;&#x2C6; G, x, y â&#x2C6;&#x2C6; F has just one open and dense orbit which is G/ZÎ&#x2DC; . Let x0 be the origin of FÎ&#x2DC; . Since G acts transitively on FÎ&#x2DC; , all the G-orbits of the diagonal action have the form G Âˇ (x0 , y), with y â&#x2C6;&#x2C6; FÎ&#x2DC;â&#x2C6;&#x2014; . Thus, the G-orbits are in bijection with the orbits through wy0 , w â&#x2C6;&#x2C6; W, where y0 is the origin of FÎ&#x2DC;â&#x2C6;&#x2014; . We obtain: Proposition 7.2.3. [16, Prop. 3.1] The orbit GÂˇ (x0 , w0 y0 ) is open and dense in FÎ&#x2DC; Ă&#x2014; FÎ&#x2DC;â&#x2C6;&#x2014; and identifies to G/ZH . Remark 7.2.4. Katzarkov, Kontsevich, and Pantev [20] give three definitions of Hodge numbers for Landauâ&#x20AC;&#x201C;Ginzburg models and conjecture their equivalence. Understanding the relation between the diamonds we presented here and those Hodge numbers provides a new perspective to our work. Acknowledgement It is a pleasure to thank Denis Auroux for suggesting corrections and improvements to the text. E. Gasparim and L. Grama were supported by Fapesp under grant numbers 2012/10179-5 and 2012/21500-9, respectively. L. A. B. San Martin was supported by CNPq grant no 304982/2013-0 and FAPESP grant no 2012/17946-1.

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

Callander, Grama, San Martin Depto. de Matemática, Imecc - Unicamp, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz. 13083-859 Campinas - SP, Brasil, briancallander@gmail.com, linograma@gmail.com, smartin@ime.unicamp.br.

Gasparim Depto. Matemáticas, Universidad Católica del Norte, Antofagasta, Chile, etgasparim@gmail.com

References [1] Abouzaid, M. ; Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Math. (N.S.) 15 (2009), no. 2, 189–270. [2] Abouzaid, M. ; Auroux, D. ; Efimov, A. ; Katzarkov, L. ; Orlov, D. ; Homological mirror symmetry for punctured spheres, J. Amer. Math. Soc. 26 (2013), 1051–1083 . [3] Abouzaid, M. ; Auroux, D. ; Katzarkov, L. ; Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfacdes, arXiv:1205:0053. [4] Abouzaid, M. ; Smith, I. ; Homological mirror symmetry for the 4-torus. Duke Math. J. 152 (2010), no. 3, 373–440. [5] Amorós, J. ; Bogomolov, F. ; Katzarkov, L. ; Pantev, T. ; Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Differential Geom. 54 (2000), no. 3, 489–545. [6] Auroux, D.; Katzarkov, L.; Orlov, D.; Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. Math. 167 (2008), 867–943. [7] Auroux, D. ; Katzarkov, L. ; Orlov, D. ; Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves, Inventiones Math. 166 (2006), 537–582. [8] Bondal, A. ; D. Orlov,D. ; Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327–344.

Symplectic Lefschetz fibrations from a Lie theoretical viewpoint

[9] Ballard, M. ; Diemer, C. ; Favero,D. ; Katzarkov, L ; Kerr, G. ; The Mori program and non-Fano homological mirror symmetry, arXix:1302.0803. [10] Callander, B. ; Lefschetz Fibrations, Master’s Thesis, Universidade Estadual de Campinas (2013). [11] Callander, B. ; Gasparim, E. ; Hodge diamonds and adjoint orbits, arXiv:1311.1265. [12] Donaldson, S. K .; Lefschetz fibrations in symplectic geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. Extra Vol. II (1998), 309–314. [13] Efimov, A. ; Homological mirrror symmetry for curves of higher genus Adv. Math. 230 (2012), no. 2, 493–530. [14] Fukaya, K. ; Mirror symmetry of abelian varieties and multi-theta functions. J. Algebraic Geom. 11 (2002), no. 3, 393–512. [15] Gasparim, E ; Grama, L ; San Martin, L. A. B. ; Lefschetz fibrations on adjoint orbits, to appear on Forum Math., arXiv:1309.4418. [16] Gasparim, E ; Grama, L ; San Martin, L. A. B. ; Adjoint orbits of semisimple Lie groups and Lagrangian submanifolds, to appear on Proc. Edinburgh Math. Soc., arXiv:1401.2418. [17] Gross, M. ; Katzarkov, L. ; Rudatt, H. ; Towards mirror symmetry for varieties of general type, arXiv:1202:4042. [18] Gompf, R. E. ; Symplectic structures from Lefschetz pencils in high dimensions, Geometry & Topology Monographs 7: Proceedings of the Casson Fest (2004) 267–290. [19] Gompf, R. ; Stipsicz, A. ; An introduction to 4-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Math. Society, Providence (1999). [20] Katzarkov, L. ; Kontsevich, M. ; Pantev, T. ; Bogomolov-TianTodorov theorems for Landau-Ginzburg models, arXiv:1409.5996. [21] Kontsevich, M. ; Homological algebra of Mirror Symmetry, Proc. International Congress of Mathematicians (Zurich, 1994) Birkhäuser, Basel (1995) 120–139.

B. Callander, E. Gasparim, L. Grama and L. A. B. San Martin

[22] Polishchuk, A. ; Zaslow, E. ; Categorial mirror symmetry: The elliptic curve, Adv. Theor. Math. Phys. 2 (1998) 443–470. [23] Seidel, P. ; Homological mirror symmetry for the genus two curve. J. Algebraic Geom. 20 (2011), no. 4, 727–769. [24] Seidel, P. ; More about vanishing cycles and mutation. Symplectic Geometry and Mirror Symmetry, World Scientific, 2001, 429–465. [25] Seidel, P. ; Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Math. Soc., Zurich (2008). [26] Sheridan, N. ; Homological Mirror Symmetry for Calabi–Yau hypersurfaces in projective space, arXiv:1111.0632. [27] Smith, I. ; Floer cohomology and pencils of quadrics. Invent. Math. 189 (2012), no. 1, 149–250.

Morfismos, Vol. 22, No. 2, 2018, pp. 21–51 Morfismos, Vol. 22, No. 2, 2018, pp. 21–51

Applications of Gauged Gromov-Witten Theory: A Survey Applications of Gauged Gromov-Witten Theory: A González Survey Eduardo Eduardo González Abstract This is a short survey article on applications of gauged GromovAbstract Witten theory into the understanding of Gromov-Witten theory of GIT quotients of a smooth projective variety of bygauged a reductive This is a short survey article on applications Gromovgroup. In particular we will explain how several classical results Witten theory into the understanding of Gromov-Witten theory in equivariant cohomology quantum cohomology. of GIT quotients of a extend smoothtoprojective variety by a These reductive include wallIncrossing results, Witten localisation and classical abelianisagroup. particular we will explain how several results tion.in We also describe a GIT version of the so called crepant equivariant cohomology extend to quantum cohomology. These conjecture. include wall crossing results, Witten localisation and abelianisation. We also describe a GIT version of the so called crepant conjecture. In memory of Samuel Gitler

2010 Mathematics SubjectInClassification: 14N35Gitler (primary), and 53D45 memory of Samuel (secondary) Keywords and phrases:Subject Gauged Gromov-Witten Theory, Crepant 2010 Mathematics Classification: 14N35 (primary), andcon53D45 jecture, Wall-Crossing, Gromov-Witten invariants, Abelianisation, GIT (secondary) quotients, equivariant quantum cohomology, quantumTheory, Kirwan Crepant map. Keywords and phrases: Gauged Gromov-Witten conjecture, Wall-Crossing, Gromov-Witten invariants, Abelianisation, GIT quotients, equivariant quantum cohomology, quantum Kirwan map.

Introduction

This1noteIntroduction is a survey, and our objective is to explain several results in the series of joint papers [22, 19, 20, 21] with Chris Woodward. This material is based several by the particular a in This note is a on survey, andtalks our given objective is toauthor, explaininseveral results series lectures given papers at CIMPA-CIMAT-SWAGP Research School atThis theofseries of joint [22, 19, 20, 21] with Chris Woodward. the material CIMAT, is México 2013) a lecture at the Second a based (Winter on several talksand given by thegiven author, in particular International School given on TQFT, Langlands and MirrorResearch Symmetry, in at series of lectures at CIMPA-CIMAT-SWAGP School Playa CarmenMéxico (Spring(Winter 2014). The present are essentially thedel CIMAT, 2013)results and awe lecture given at the Second extensions of well known results in equivariant cohomology into in International School classical on TQFT, Langlands and Mirror Symmetry, quantum and as such we The will results use thewe term quantisation in Playa cohomology, del Carmen (Spring 2014). present are essentially extensions of well known classical results in equivariant cohomology into quantum cohomology, and as such 21 we will use the term quantisation in 21

Eduardo González

this sense. In Section 2, we begin by revising classical tools relating the cohomology of a GIT quotient X G and the equivariant cohomology HG (X), chiefly to introduce notation. We then discuss Kalkman’s wall crossing formula, Martin’s abelianisation formula and Witten’s localisation, which we are the results we are quantising. These results can be understood in terms of trace maps (integration), the Kirwan map and certain fixed point traces, twisted by Euler classes corresponding to fixed point components. We will see that the same results will hold in quantum cohomology, provided that we appropriately redefine these main components. The traces are replaced by appropriately introducing higher degree maps and defining generating functions or potentials of gauged and usual Gromov-Witten invariants. The Kirwan map is replaced by Woodward’s quantum analogue and the fixed point contributions will arise as potentials twisted by Euler classes of index bundles associated to fixed points. Due to the nature of the paper, we will only focus on explaining the results and we will do some basic examples. In Section 3 we discuss the construction of the moduli space of Mundet semi-stable gauged maps and the definition of the gauged GromovWitten potential. Here we emphasise that gauged maps are equivariant lifts of maps into X G, and to recover the maps into the quotient one needs to consider limits in the semi-stability condition. In Section 4.1 we proceed to explain wall crossing for gauged Gromov-Witten potentials as well as its descent (adiabatic limit) to Gromov-Witten potentials of X G. We then explain in Section 4.4 how this can be used to show that the Gromov-Witten potentials of two GIT quotients related by a crepant birrational map are essentially the same. In Section 4.5 we describe how gauged Gromov-Witten theory can be used to quantise Witten’s localisation and then use it to show abelianisation formulas, that is, to relate the Gromov-Witten theories of X G and that of the quotient X T by its maximal torus.

Classical equivariant cohomology results

We follow the same set up as in [34]. Let G be a reductive group and consider a non-singular polarised G variety (X, L), that is an ample line bundle L → X equipped with a linearisation (lift) of the action. We let X ss denote the semistable locus, that is the subset of points x ∈ X such that s(x) = 0 for some invariant section s ∈ H 0 (X, L⊗n )G and some integer n. We will assume in most cases, unless explicitly stated,

Applications of Gauged Gromov-Witten Theory: A Survey

that all semi-stable elements are strictly stable (stable=semistable); in such case G acts with finite stabilisers on the semistable set, so that the GIT quotient X G := X ss /G, is a Deligne-Mumford stack. In this paper we will assume that the quotient is actually smooth, to simplify the exposition, however the results hold assuming that the quotient has a projective coarse moduli. Let HG (X; Q) denote the equivariant cohomology, which we will also identify with the cohomology of the quotient stack X/G. In symplectic geometry, Kempf-Ness [31] identified the GIT quotient as the symplectic quotient: the quotient of the zerolevel set of the moment map by the maximal compact in G. Let κX,G : HG (X; Q) → H(X G) denote the Kirwan map, given by restriction to the semistable locus and then descent to the quotient. Integration over X G defines a trace (1) (Trace) τX G : H(X G) → Q, α → α. [X G]

We now review classical equivariant cohomology results.

2.1

Wall crossing and Kalkman’s formula

The GIT quotient X G depends on L, or equivalently, a choice of moment map. The dependency on the variation of L is studied in [17, 7, 12, 41]. Under suitable stable=semistable conditions, X G undergoes a sequence of weighted blow-ups and blow-downs. The class of birational equivalences which appear via variation of GIT is quite large. In fact, for the so-called Mori dream spaces, any birational equivalence can be written as a composition of birational equivalences induced by variation of GIT. We are interested in what happens at the level of intersection parings in cohomology, and in GW theory correlators. The following material is well known, however we need it to introduce notation to explain our results. Kalkman [27] studied the question of how the cohomology of the quotient depends on the polarisation, and provides a wall-crossing formula for the intersection pairings. Let X ± G denote the associated GIT quotients corresponding to two polarisations L± → X, and let κG X,± : HG (X) → H(X ± G) be the Kirwan maps and let τX ± G : H(X ± G) → Q denote integration over X ± G. Kalkman’s formula expresses the difference between the integrals τX ± G ◦ κG X,± as a sum of fixed point contributions for the integral of a class α ∈ HG (X) over

Eduardo González

X. In other words, it measures the failure of the following square to commute, κG X,+

HG (X)

H(X + G)

H(X − G)

τX + G

G κX,−

τX − G

by an explicit sum of wall-crossing terms. To explain properly what (1−t)/2 ⊗ these contributions are, consider the interpolation Lt := L− (1+t)/2 L+ for rational t ∈ (−1, 1). The family of GIT quotients X Lt G given by the variation of semi-stability is recovered by the master space [41], the quotient M = P(L+ ⊕ L− ) G. M itself has a C× action given by the scaling on the fibres of M . There is a natural linearisation of this action, induced by O(1) → P(L+ ⊕ L− ), which in turn produces a family of quotients M t C× by considering the semi-stability with respect to O(t). The main result of variation of GIT is that under appropriate stable=semi-stable conditions M t C× is naturally identified with X Lt G. The fixed point set of the C× -action on M is given as follows. For any ζ ∈ g, we denote by Gζ ⊂ G the stabilizer of ζ under the adjoint action of G. Let T ⊂ G be a maximal torus and t ⊂ g the corresponding Cartan algebra, and W = N (T )/T its Weyl group. For any ζ ∈ t, we denote by Wζ resp. WCζ the group of w ∈ W that fix the element ζ ∈ g resp. line Cζ ⊂ g. Thus the quotient WCζ /Wζ is either isomorphic to {±1} or to {1}, depending on whether or not there is a Weyl group element acting as −1 on Cζ. Suppose that stable=semistable for the G-action on P(L− ⊕ L+ ), so that M is a smooth × proper Deligne-Mumford stack with C× action. Any x ∈ M C with x = [l] for some l ∈ P(L− ⊕ L+ ) has the property that for all z ∈ C× , zl = z ζ l for a unique ζ ∈ g. For each ζ ∈ t there is a morphism C× with fiber W /W . The images of ι cover ιζ : X ζ t (Gζ /C× Cζ ζ ζ ζ )→M ×

M C , disjointly after passing to equivalence classes of one-parameter (α)|M C× under ιζ is subgroups. For any α ∈ HG (X), the pull-back of κ equal to image of α under the restriction map HG (X) → HC× (X ζ t (Gζ /C× ζ )). ζ

The pull-back of the normal bundle NM C× of M C under ιζ is canonically isomorphic to the image of NX ζ /(g/Rζ) under the quotient map

Applications of Gauged Gromov-Witten Theory: A Survey

X ζ → X ζ (Gζ /C× ζ ), by an isomorphism that intertwines the action of × × Cζ on (NX ζ /(g/Rζ)) (Gζ /C× ζ ) with the action of C on NM C× . The × group Cζ ⊂ Gζ acts trivially on X ζ , which is therefore a Gζ /C× ζ -space. This is the “smaller structure group” acting on the wall terms. For any fixed point component X ζ,t ⊂ X ζ that is t-semistable, we denote by νX ζ,t the normal bundle of X ζ,t modulo g/Cζ, quotiented by Gζ /C× ζ . × ζ,t ζ,t Let jζ,t : HG (X) → HC× (X ) → H(X (Gζ /Cζ )), then we define ζ

(2)

ζ (Fixed Point Trace)τX,ζ,t : HG (X) → Q[ξ, ξ −1 ], jζ,t (α) ∪ EulC× (νX ζ,t )−1 α →

[X ζ,t (Gζ /C× ζ )]

where ξ is the equivariant parameter for C× ζ . Therefore, with the considerations above (stable=semistable for the G action on P(L− ⊕L+ )) we have (3)

(Kalkman’s wall-crossing formula) τX − G ◦ κG X,− =

t∈(−1,1),[ζ]

τX + G ◦ κG X,+ −

|Wζ | Gζ Residξ τX,ζ,t , |WCζ |

where the sum is over one-parameter subgroups [ζ] of G, up to conjugacy. The formula (3) holds more generally, e.g. for certain quasiprojective varieties, such as vector spaces whose weights are contained in an open half-space. Example 2.1.1. Let us exemplify the notation above. Let G = C× act on X = CN by scalar multiplication, so that HG (X) = Q[ξ]. Let L± correspond to the weights ±1, thus X − G is empty and X + G = PN −1 . There is a unique singular value t = 0, corresponding to the origin 2 0 ∈ X. κG X : HG (X) → H(X G) sends ξ ∈ HG (X) to the hyperplane 2 class h ∈ H (X G). The integrals PN −1 ha for a ∈ Z≥0 can be computed via wall-crossing. For the empty side, the integral is zero. By the Kalkman formula (3) ha = Resξ ξ a ∪ EulG (CN )−1 = Resξ ξ a /ξ N N −1 P [0] 1 a=N −1 = 0 otherwise showing that hN −1 is the dual of the fundamental class.

2.2

Eduardo González

Witten localisation and Abelianisation

It is natural to ask if we can compute the composition τX G ◦ κX,G exclusively in terms of the G-equivariant cohomology on X, for instance, to compute the cohomology of X G in terms of the G-equivariant cohomology of X. Witten [42] introduced non-abelian localisation to compute τX G ◦ κX,G in terms of a trace map (4)

(Witten’s Trace)

G τX

: HG (X) → Q,

α →

α,

[X]×k

which is given by integration over X and a unitary form k of the Lie algebra g. The integral of a polynomial over k may be defined via various regularisation procedures, see [38, 37, 44], which we will not explain, since they are unnecessary for the quantised version in Section 4.2.1. Our original motivation for the quantum version of Witten’s localisation was the quantum Martin conjecture of Bertram et al [4] which compares Gromov-Witten invariants of a GIT X G and the quotient X T by a maximal torus T ⊂ G, as we will see in Section 4.5. Let us first discuss classical abelianisation or Martin’s formula [33]. T acts on (X, L) as well, and it is not hard to see that the semistable locus of the G action X ss (G) lies in X ss (T ) so that there is a natural map X ss (G)/T → X ss (T )/T = X T and a quotient X ss (G)/T → X ss (G)/G = X G by the residual action. The relation of the two GIT quotients is given by νg/t , the bundle over X T induced from the trivial bundle with fibre g/t over X. We let τX T : H(X T ) → Q denote the Eul(νg/t )-twisted integration map, τX T : H(X T ) → Q,

α →

[X T ]

α ∪ Eul(νg/t ).

Let W = N (T )/T denote the Weyl group of T ⊂ G and RestrG T : HG (X) → HT (X) the restriction map, which induces an isomorphism HG (X) → HT (X)W . Suppose that stable=semistable for the actions of T and G on X. Then integration over X G and X T are related by (5)

(Martin’s Abelianisation formula)

τX G ◦ κX,G =

|W |−1 τX T ◦ κX,T ◦ RestrG T .

Applications of Gauged Gromov-Witten Theory: A Survey

In other words, the following diagram commutes: HG (X, Q) ∼ = HT (X, Q)W κX,G

H(X G, Q)

κX,T

H(X T, Q)

τX G

|W |−1 τX T

Moreover Martin proved that there exists a surjective map µG T : H(X T ) → H(X G)

(6)

whose kernel is the cup product with Eul(g/t). Given classes αG ∈ H(X G), αT ∈ H(X T ), we have αG = µG T ◦ αT iff αG ∪ κX,G (β) = |W |−1 αT ∪ κX,T (β) ∪ Eul(g/t) [X G]

[X T ]

for all β ∈ HG (X) ∼ = HT (X)W .

Gauged Gromov-Witten theory

In order to extend the results explained in the previous section, we first need to introduce gauged maps as equivariant lifts of stable maps. Let C be a smooth connected projective curve. Since X G is embedded in the quotient stack X/G, a map C → X G is in principle a gauged map: a morphism C → X/G, or more explicitly, a pair (P, u) of a principal G-bundle P → C over C and a section u of the associated bundle P (X) := P ×G X → C. In this way, gauged maps are naturally the algebraic analogues of symplectic vortices [8, 9, 8, 18] similar to the gauged σ-models of Witten [43]. Therefore the space of gauged maps is a natural extension of both, the space of maps Maps(C, X) and the stack BG of principal bundles. Since we are interested in invariants, we will also consider the moduli of maps with n different markings, c = (c1 , · · · , cn ) ∈ C n . We consider the stack of n-pointed nodal gauged G maps Mn (C, X). These are pairs (u, c) of a stable n-pointed map u :

Eduardo González

→ C × X/G such that the projection onto the first factor C has class C [C]. The degree of a gauged map is its push forward in H2 (X/G) = H2G (X). Here, we identify the (co)homology of the quotient stack X/G with the equivariant (co)homology of X [13]. Thus for a gauged map, 0 of the the principal bundle lies exclusively on a principal component C isomorphic to C, and the principal bundle over all other nodal curve C, map to a point. This means that the nodal map defines components of C → P (X) with base class [C], that is, the composition a stable map u : C of u with the projection P (X) → C has class [C]. This means that nodal gauged maps are allowed to acquire “fibre bubbles”, rational bubble trees attached on the fibres. The underlying curve in the gauged map lies in the Fulton-McPherson compactification Mn (C) of the space of maps, rather than the Deligne-Mumford compactification as in usual GW theory, since it has a principal component. This compactification is the space Mg(C),n (C, [C]) of genus g(C) stable maps into C, of fixed G,st

degree [C] ∈ H2 (C). We let Mn (C, X) denote the substack of gauged → P (X) is a stable map in the usual sense maps for which (u, c) : C G G,st of Kontsevich. In general Mn (C, X), Mn (C, X), MG n (C, X) are nonfinite type, non-separated Artin stacks [45, Theorem 5.2]. Since we are interested in invariants, we need to impose a semi-stability condition to guarantee a good moduli space supporting a perfect obstruction theory. We want to differentiate the construction above with equivariant GW theory, as introduced by Givental [15]. In this case, the stable maps are requested to take values in the fibres of the homotopy quotient X ×G EG → BG (the geometric realisation of the stack X/G), that is, they only lie in the X direction.

3.1

Mundet semi-stability and gauged GW invariants

A choice of polarisation L → X of X (equipped with a compatible lift of the G-action on L) gives a semi-stability condition for gauged maps, as introduced by Mundet i Riera [35, 36]. Mundet’s semi-stability couples Ramanathan’s semi-stability for principal bundles and the HilbertMumford semi-stability for X. Suppose (P, u) : C → X/G is a gauged map. Let R denote a parabolic subgroup of G, and let σ : C → P/R denote a reduction of structure group. Let λ be a co-weight of R (we identify g∼ =g∨ ) and consider the one-parameter subgroup C× → R given by z → z −λ = exp(− ln(z)λ). The pair (λ, σ) yields the associated graded, a pair (Gr(P ), Gr(u)) consisting of a bundle Gr(P ) whose structure group → reduces to the Levi subgroup of P and a stable section Gr(u) : C

Applications of Gauged Gromov-Witten Theory: A Survey

If R = LU is a Levy decomposition, Gr(P )(X) from a nodal curve C. ∗ then Gr(P ) = i∗ p∗ σ P , where p : R → R/U = L, i : L → G are the natural maps. For our purposes we prefer the presentation via degeneration which we will describe in what follows. For each z ∈ C× , we consider the G bundle P(z,σ) = σ ∗ P ×R,z −λ G where the action on G is by conjugation. The limit as z → 0 exists, if λ is dominant, and agrees with Gr(P ). By λ dominant we mean that it is zero on the connected component of the centre of G and it is positive on the roots of T . The twisted map z −λ u is a section of the associated bundle P(z,σ) (X), and its limit as z → 0 converges in the Gromov sense to a nodal section of Gr(P )(X). This is the graded section Gr(u). Over the principal Gr(u) takes values in the fixed point set X λ of the component C0 ∼ =C, automorphism induced by λ, and so has a well-defined Hilbert-Mumford weight µHM (σ, λ) determined by the polarisation L, given as the usual Hilbert-Mumford weight [34, Section 2] at a generic point in C0 . The Ramanathan weight (cf. [2, Definition 3.2], [39, 40]) µR (σ, λ) of (P, u), with respect to (σ, λ) is given by the number of the line first Chern ∗ bundle determined by λ: µR (σ, λ) = [C0 ] c1 (σ P ×R Cλ ). Definition 3.1.1. The Mundet weight is then the coupling (7)

µM (σ, λ) := µHM (σ, λ) + µR (σ, λ).

The map (P, u) is Mundet semistable if µM (σ, λ) ≤ 0 for all pairs (σ, λ) for λ dominant, and Mundet stable if the above inequalities are satisfied strictly. This definition carries to stable gauged maps, since there is no contribution to the weight coming from the bubbles. We denote by G Mn (C, X) the space of semi-stable nodal pointed gauged maps. Remark 3.1.2. The Hilbert-Mumford weight can be computed in terms of the Moment map, using the usual correspondence between linearisations of actions and moment maps [28]. Let ρ : X → P(V ), V = H 0 (X, L)∨ the embedding given by the (very ample) polarisation L, so that G → GL(V ). Let Φ = ρ ◦ ΦP : X → P(V ) → gR denote the restriction to X of the moment map ΦP of P(V ). In this case the Hilbert-Mumford weight is just given by (8) µHM (σ, λ) = P (Φ) ◦ Gr(u), λ , C0

where P (Φ) is the map Φ induces on P (X), and the integral is over the principal component.

Eduardo González

Example 3.1.3. So far the definition of semi-stability was done in the projective setting. If X is affine, we can still define semi-stability in the same exact way, but only for the pairs (σ, λ) for which the graded (Gr(P ), Gr(u)) exists. To exemplify this, consider the diagonal action of G = C× in the affine space X = CN , where we take the moment map to be Φ : CN → gR given by z → −|z|2 + 1. Let C = P and P → P be a bundle of degree d ∈ Z. Then, R = G and let λ ∈ Z, denote a coweight of R. Since all the reductions σ are trivial we have Gr(P ) = P . The associated bundle P (X) = P ×C× CN = O(d)N → P has sections only if d ≥ 0. We need to determine which sections u ∈ H 0 (P, O(d)N ) are semi-stable. For all λ ∈ Z the limit Gr(u) = limz→0 z −λ u exists and is zero if u ≡ 0; it does not exist if λ > 0 and u = 0; and it exists and is zero if λ < 0 and u = 0. Note that since X is affine, there is no fibre bubbling, so Gr(u) ≡ 0, whenever the limit above exists. Thus, the weight (7) is given by µM (σ, λ) = d, λ + Φ(Gr(u)), λ VolP = d, λ + λ, P

using that Φ(Gr(u)) = 1 and normalising so that Vol(P) = 1. Therefore, the semi-stable sections are those non-zero, and the unstable one is u ≡ 0. Therefore MG (C, X) = (H 0 (P, O(d)N ) \ {0})/C×= P(H 0 (P, O(d)N )) = PN (d+1)−1 . Since there is no bubbling, G

Mn (C, X) = MG (C, X) × Mn (C) = PN (d+1)−1 × Mn (C). Example 3.1.4. We can generalise the previous example when C = P. Consider the toric action of a torus G = T = (C× )k on the affine space CN with the weights µ1 , . . . , µN ∈ t∨ . The semi-stability condition is determined by a choice of character χ ∈ t∨ , giving a linearisation on the trivial bundle Lχ . In this case the semistable set is given by X ss = {(z1 , . . . , zN ) ∈ CN : χ ∈ span{µi , zi = 0}}, and X χ T is T a toric variety. The moduli space M (P, X, d) can be computed as follows. A bundle P with degree d ∈ H2T (X) = tZ has an associated bundle P ×T X which is just ⊕i OP (di ), where di = d, µi ∈ Z and thus H 0 (P ×T X) is identified with X(d) := ⊕i Cmax(0,di +1) , where T acts with weight µi repeated max(0, di +1). By rescaling the linearisation Ltχ , and assuming t large enough so that the Mundet weight (7) is dominated by

Applications of Gauged Gromov-Witten Theory: A Survey

the Hilbert-Mumford term, we obtain M T (P, X, d) = X(d) Ď&#x2021; T which is itself a toric variety. These spaces are the so called quasi-maps as it was defined by Givental in [15]. Fix a degree class d â&#x2C6;&#x2C6; H2G (X), and assume that stable=semistable for all gauged maps of degree d, and denote the substack of degree G d by Mn (C, X, d). Similar to GW theory, there is an evaluation at G the markings taking values in the quotient stack, ev : Mn (C, X, d) â&#x2020;&#x2019; G (X/G)n , and a forgetful morphism f t : Mn (C, X, d) â&#x2020;&#x2019; Mn (C). The following result is a summary of [19, Proposition 2.1.3], [46, Proposition 5.12, Theorem 5.14, Example 6.6]. Theorem 3.1.5. Assume stable=semistable for gauged maps, then the G moduli stack Mn (C, X, d) is a proper, separated Deligne-Mumford stack of finite type with a perfect obstruction theory. The properness portion of the result uses a Hitchin-Kobayashi correspondence with the moduli of vortices, see [46, Section 5] for a discussion as well as relations to other compactifications. The perfect obstruction theory is similar to the construction in GW theory. There are universal G G G maps p : U n (C, X) â&#x2020;&#x2019; Mn (C, X), ev : U n (C, X) â&#x2020;&#x2019; X/G. And thus we can take the complex Rpâ&#x2C6;&#x2014; evâ&#x2C6;&#x2014; T (X/G)â&#x2C6;¨ as the perfect obstruction theory, yielding a virtual fundamental class [5]. Define gauged GW invariants as â&#x160;&#x2014;n Âˇ; Âˇ G â&#x160;&#x2014; H(Mn (C)) â&#x2020;&#x2019; Q ; n,d : HG (X) Îą, Î˛ dg,n

G [Mn (C,X,d)]vir

evâ&#x2C6;&#x2014; Îą â&#x2C6;Ş f tâ&#x2C6;&#x2014; Î˛,

by using integration over the virtual fundamental class. Example 3.1.6. Let us continue Example 3.1.3 above, with G = CĂ&#x2014; , C = P, X = CN so that X G = PN â&#x2C6;&#x2019;1 . In this case H â&#x2C6;&#x2014; (X/G) = Q[Îž], where Îž is the equivariant parameter. Let n = 3, be the number of markings. It is not hard to see that the pull back of Îž under (any of) the evaluation map is the hyperplane class h â&#x2C6;&#x2C6; H 2 (PN (d+1)â&#x2C6;&#x2019;1 ). Then M3 (C, X, d) = PN (d+1)â&#x2C6;&#x2019;1 Ă&#x2014;M3 (P). The space M3 (P) is six dimensional, and we take Î˛ â&#x2C6;&#x2C6; H 6 (M3 (P)) the point class, which corresponds to fixing the markings. Thus 1 if a + b + c = N (d + 1) â&#x2C6;&#x2019; 1; Îž a , Îž b , Îž c ; Î˛ G h a hb hc = n,d = 0 otherwise. PN (d+1)â&#x2C6;&#x2019;1

Eduardo GonzaĚ lez

Note that this does not quite agree with the 3-point GW invariant N â&#x2C6;&#x2019;1 . For instance if N = 2 the Gromov-Witten invariant ha , hb , hc Pd 1 a b c P h , h , h d = 1 only if d = 1 and a+b+c = 3 or d = 0 and a+b+c = 1. Here we also denote by h the hyperplane class h â&#x2C6;&#x2C6; H 2 (P1 ), which is the image of the generator Îž under the classical Kirwan map. This discrepancy is corrected of the quantum Kirwan map. Remark 3.1.7. As a side remark, recall that GW invariants define a cohomological field theory (CohFT) in the sense introduced by Kontsevich and Manin [30]. In a similar fashion gauged GW invariants define a CohFT trace, since its underlying combinatorial structure is controlled by parametrised curves, the Fulton-McPherson compactification of curves. We will not need this here, and the interested reader can consult [45, Section 2.2] for more details. Let (9) Î&#x203A;G X :=

ďŁą ďŁ˛ ďŁł

dâ&#x2C6;&#x2C6;H2G (X;Q)

ďŁź ďŁ˝ cd q d : â&#x2C6;&#x20AC;e > 0 # d : cG (L), d < e < â&#x2C6;&#x17E; ; 1 ďŁž

QHG (X) = HG (X, Q) â&#x160;&#x2014; Î&#x203A;G X

denote the equivariant Novikov ring, and the G-equivariant quantum cohomology of X respectively, equipped with the usual quantum product using the G-equivariant GW invariants of X taking values in H(BG). Such invariants are defined as the usual invariants, but by equivariant integration over the moduli of maps with respect to the inherited action of G [15]. For the applications we want to discuss, it is convenient to introduce the gauged potential as the map (10)

(Gauged GW potential) G Ď&#x201E;X,L (Îą) :=

nâ&#x2030;Ľ0 dâ&#x2C6;&#x2C6;H G (X;Q)

G Ď&#x201E;X,L : HG (X) â&#x2020;&#x2019; Î&#x203A;G X;

Îą, . . . , Îą; 1 G n,d

qd , n!

and extend to QHG (X) by linearity. We drop L from the notation whenever there is no risk of confusion.

Quantised Results

We are now ready to discuss the main applications.

Applications of Gauged Gromov-Witten Theory: A Survey

4.1

Wall crossing for gauged maps.

Similar to the considerations in Section 2.1, we assume that X is a projective G-variety with two given polarisations L± → X. The variation of semi-stability for gauged maps with respect to the interpolation (t+1)/2 (t−1)/2 ⊗L− , t ∈ (−1, 1)∩Q yields the family of moduli spaces L t = L+ G

Mn (C, X, d, Lt ) of Lt -semistable gauged maps. The relation between the associated gauged potentials is given by a formula similar to the classical wall-crossing Equation (3). We follow a similar proof as well. We construct a master space M̃ equipped with a C× action, realising G each Mn (C, X, d, Lt ) as a quotient M̃ t C× . Then the result will follow by applying virtual localisation [23] to the master space. Virtual Kalkman in this context is due to Kiem-Li [29], under the same hypotheses G imposed in [23], i.e. one needs to verify that the stacks Mn (C, X, d, L± ) are embedded in smooth Deligne-Mumford stacks. For more details the reader can consult [21, Theorem 2.6] and the references therein. From this discussion we have. Theorem 4.1.1 (Theorem 3.8 [21]). The difference of the gauged potentials is given by (11) G G (Wall crossing for gauged potentials) τX,+ (α) − τX,− (α) =

Res τX,ζ,Lt (α).

Here the sum of contributions on the right range over one-parameter subgroups, and the terms are residues of the fixed point potentials τX,Gζ ,Lt corresponding to the subgroup. This potential is constructed as follows. For each ζ ∈ g and fixed point component X ζ,t ⊂ X ζ that is G t-semistable for some t ∈ (−1, 1), let Mn ζ (P, X, L± , ζ, t, d) denote the stack of Mundet semistable morphisms P → X/Gζ that are C× ζ -fixed and take values in X ζ,t on the principal component. Note that the bubbles can take values in the whole of X. Let Ind(T (X/G))+ ⊂ Ind T (X/G)) denote the moving part of the normal complex Ind(T (X/G)) with respect to the action induced by ζ, considered as an object in the derived category of bounded complexes of coherent sheaves on G

Mn ζ (P, X, L± , ζ, t, d).

Eduardo González

Virtual integration over the stacks Mn ζ (P, X, L± , ζ, t, d) defines a “fixed point contribution” G

ζ −1 τX,ζ,t : QHG (X) → ΛG X [ξ, ξ ],

which sends α into G

ζ d,n≥0 [Mn (P,X,L± ,ζ,t,d)]

ev∗ (α, . . . , α) ∪ EulC× (Ind(T (X/G))+ )−1 ζ

qd , n!

where again HC× (pt) = Q[ξ]. ζ

The wall crossing formula (11) can be thought of a generalisation of the wall-crossing formula for abelian vortices studied in [11]. In order to prove a wall crossing formula for the GW invariants of the quotient, we need to take an adiabatic limit in the polarisation.

4.2

Adiabatic limits.

Adiabatic limits appear as special cases of the wall crossing formula (11), when the polarisations are just a rescaling of each other, L+ = Lt− , L = L− , so that the variation of semi-stability depends only on the rational t ∈ (0, ∞). In the symplectic category, this is equivalent to the dependence on the “area form”, as we originally discussed it in [19]. 4.2.1

Small area limit, t → 0.

The moduli stack of Mundet semistable gauged maps in this limit is identified with a quotient of the moduli stack of parametrized stable maps to X. Let C ∼ = P be a curve of genus zero. A stable map u = (uC , uX ) : Ĉ → C × X of degree (1, d), d ∈ H2 (X) is zero-semistable ss if and only if uC (u−1 X (X )) is dense in C, that is, if it is generically semi-stable. Denote by Mn (C, X, d)0-ss ⊂ Mn (C, X, d) := M0,n (C × X, (1, d)) the 0-semistable locus and by (12)

Mn (C, X, d) 0 G = Mn (C, X, d)0-ss /G

the quotient stack, of the G action given by post-composition. Theorem 4.2.1. [22, Theorem 1.2] With the considerations above, let d ∈ H2 (X; Z) ⊂ H2G (X, Z). There exists a t0 such that for t < t0 , there G is an isomorphism Mn (C, X, d) 0 G → Mn (C, X, d; Lt ) of DeligneMumford stacks equipped with perfect relative obstruction theories.

Applications of Gauged Gromov-Witten Theory: A Survey

The proof relies on the following. The rescaling of the polarisation L → Lt has a linear effect on the Mumford term in the Mundet’s weight (7): µM (σ, λ) = tµHM (σ, λ) + µR (σ, λ). For t sufficiently small, Mundet semi-stability is thus equivalent to semi-stability of the G-bundle and therefore semi-stability of a gauged map reduces to that of the bundle. Since C is rational this is equivalent for the bundle P to be the trivial. G The relative obstruction theory on Mn (C, X) has complex (Rp∗ e∗ T (X/G))∨ given by descent from (g → T Mn (C, X))∨ , since every t-semistable map has underlying trivial bundle. The latter is isomorphic to the relative obstruction theory on Mn (C, X) G. For abelian actions on affine spaces, this limit is related to the toric map spaces of Givental [16, Section 5]. An analogous formula as in the potential (16), but integrating over the moduli spaces Mn (P × X, (1, d)) G yields a (t = 0) potential (13)

(Quantum Witten Trace)

G τX : QHG (X) → ΛG X

which is called in [19] the quantum Witten trace, since when d = 0, it recovers the trace Witten suggested for his localisation (4). Combining with Theorem 4.2.1 we obtain: (14)

G G lim τX,L = τX . t

t→0

In this “small area” limit, we can express the gauged potential in terms of morphisms from stable map spaces, at least in the case when C has genus zero. Let κM,G : H G (Mn (C, X, d) → H(Mn (C, X, d) G); τM G : H(Mn (C, X, d) G) → Q

denote the Kirwan map (that is, restriction to the 0-semistable locus G is the the and descent) and virtual integration respectively. Then τX composition of pull-back with integration over the moduli space of stable maps G : HG (X) → ΛG α → (q d /n!)τM G ◦ κM,G ev∗ (α, . . . , α). τX X, d,n

For the purposes of the next section, we remark that if C ∼ =P is × equipped with a C -action then the same results hold for the C× equivariant potentials.

4.2.2

Eduardo GonzaĚ lez

Large area limit, t â&#x2020;&#x2019; â&#x2C6;&#x17E; and the quantisation of the Kirwan map.

This case has been treated by Woodward [45, 46, 47] in the algebraic case, and by Ziltener [49] in the symplectic case, based on earlier work of Gaio-Salamon [18]. In this limit, the bundle contribution of the Mundet weight (7) vanishes, so that gauged maps take values in the semi-stable quotient X ss /G = X G. One needs to add some other corrections accounting for special affine gauged bubbles which arise in this case, contrasting to the small limit case where such special bubbling in the limit does not appear. More precisely, in this limit the gauged potential (10) factorises [45, Theorem 1.5] as (15) G (Adiabatic limit Factorisation) lim Ď&#x201E;X (Îą) = Ď&#x201E;X G â&#x2014;Ś ÎşG X (Îą), Î´â&#x2020;&#x2019;â&#x2C6;&#x17E;

that is, the diagram ÎşG X

QHG (X)

G Ď&#x201E;X

Î&#x203A;G X

QH(X G) Ď&#x201E;X G

commutes in the limit. Here the map (16) (Graph Potential) Ď&#x201E;X G (Îą) :=

Ď&#x201E;X G : QH(X G) â&#x2020;&#x2019; Î&#x203A;G X; qd â&#x2C6;&#x2014; ev (Îą, . . . , Îą) n! Mn (P,X G,d)

nâ&#x2030;Ľ0 dâ&#x2C6;&#x2C6;H2 (X G;Q)

is the graph potential that counts parametrised maps into X G in the class d and the special bubbling in the limit is encoded by the quantum Kirwan map ÎşG X . This map is defined by virtual enumeration of affine gauged maps: pairs (u, Îť) where u is a representable morphisms u : P(1, r) â&#x2020;&#x2019; X/G from a weighted projective line P(1, r), r > 0 to the quotient stack X/G, mapping the stacky point at infinity P(r) â&#x160;&#x201A; P(1, r) to the semistable locus X G â&#x160;&#x201A; X/G. These are the algebrogeometric analogues of the vortex bubbles considered in Gaio-Salamon [18]. Îť is a scaling, a meromorphic 2-form Îť : T â&#x2C6;¨ P â&#x2020;&#x2019; O(2â&#x2C6;&#x17E;) with G double pole at â&#x2C6;&#x17E;. The compactified moduli stack Mn,1 (A, X, d) of affine gauged maps of homology class d â&#x2C6;&#x2C6; H2G (X, Q) is, if stable=semistable for the G action on X, a proper smooth Deligne-Mumford stack

Applications of Gauged Gromov-Witten Theory: A Survey

with a perfect relative obstruction theory over the complexification of Stasheffâ&#x20AC;&#x2122;s multiplihedron: the moduli Mn,1 (A) of n-marked scaled lines (with a single scaling). It is equipped with evaluation maps ev Ă&#x2014; evâ&#x2C6;&#x17E; : G Mn,1 (A, X) â&#x2020;&#x2019; (X/G)n Ă&#x2014; (X G). The quantum Kirwan map is the formal morphism ÎşG X : QHG (X) â&#x2020;&#x2019; QH(X G) ; qd â&#x2C6;&#x2014; . ev ev (Îą, . . . , Îą) (Îą) := ÎşG â&#x2C6;&#x17E;,â&#x2C6;&#x2014; X n!

(17) (Quantum Kirwan map)

nâ&#x2030;Ľ0,d

It is in principle non-linear, and it is a morphism of CohFT theories as detailed in [45, Section 2.3], [47, Theorem 8.6] and WoodwardZiltener [48]. Moreover, each of its linearisations DÎą ÎşG X : TÎą QHG (X) â&#x2020;&#x2019; TÎş(Îą) QH(X G) is a homomorphism with respect to the quantum product. We are now ready to apply the wall crossing formula in several contexts. Example 4.2.2. Let us analyse the toric case. Let X = CN be the dimensional complex vector space with an action of a complex torus T = (CĂ&#x2014; )k , with weights Âľ1 , . . . , ÂľN contained in an open half-space and equipped with a polarisation so that the quotient X T is a DeligneMumford stack. First we analyse the affine gauged maps appearing in the definition of ÎşTX : An affine gauged map to X/T of homology class d â&#x2C6;&#x2C6; H2T (X, Q) is equivalent to a morphism u = (u1 , . . . , uk ) : A â&#x2020;&#x2019; X d such that the degree of uj is at most dj = Âľj , d . Let uj (â&#x2C6;&#x17E;) = uj j /(dj !) if dj â&#x2030;Ľ 0 and uj (â&#x2C6;&#x17E;) = 0 otherwise. Then, let u(â&#x2C6;&#x17E;) = (uj (â&#x2C6;&#x17E;))kj=1 denote the vector of leading order coefficients with integer exponents, then T u(â&#x2C6;&#x17E;) â&#x2C6;&#x2C6; X ss . Thus M1,1 (A, X, d) is the space of such morphisms with one finite marking and the marking at infinity up to the action of T . We T let M1,0 (A, X, d) denote the space of morphisms with only the marking at infinity up to automorphisms. In this case the automorphisms do not only come from the action of T , but also from the translation of the domain, since there is no finite marking. If T = CĂ&#x2014; acts on X = CN by T scalar multiplication then M1,1 (A, X, d) consists of tuples of polynomials of degree at most d up to the action of T , such that at least one of T the polynomials is of degree exactly d. One sees that M1,1 (A, X, d) is a vector bundle over PN â&#x2C6;&#x2019;1 of rank dN . Now, let G = (CĂ&#x2014; )N be the big torus acting in the standard way in X and consider the equivariant cohomologies QHT (X)â&#x2C6;ź = Sym(tâ&#x2C6;¨ ) â&#x160;&#x2014;

Eduardo GonzaĚ lez

Î&#x203A;TX , QHG (X)â&#x2C6;ź = Sym(gâ&#x2C6;¨ ) â&#x160;&#x2014; Î&#x203A;G X . By taking coordinates v1 , . . . vN on g we get that QHG (X) = Q[v1 , . . . , vN ] â&#x160;&#x2014; Î&#x203A;G X . Let p : QHG (X) â&#x2020;&#x2019; QHT (X) be the restriction map and let D(Âľi ) â&#x2C6;&#x2C6; H 2 (X T ) denote the divisor classes defined by vi . Then, the degree one portion ÎşG,1 X of the Kirwan map is given by [45, Lemma 8.8] (see [20] for the stacky case) ďŁŤ

ďŁ ÎşT,1 X

Âľj (d)â&#x2030;Ľ0

ďŁś

p(vj )Âľj (d) ďŁ¸ = q d

D(vj )â&#x2C6;&#x2019;Âľj (d) + higher order terms,

Âľj (d)â&#x2030;¤0

where d â&#x2C6;&#x2C6; H2T (X, Z) is a lift of an effective class. This yields the Batyrev quantum relations for the quotient X T . The surjectivity of the quantum Kirwan map for toric stacks follows similarly. The reader can consult [20] for more details, as well as the computation of the quantum cohomology of toric stacks with projective coarse moduli space. 4.2.3

Relation to the I and J functions

We continue with the same notations T = (CĂ&#x2014; )k , acting on CN . Woodwardâ&#x20AC;&#x2122;s adiabatic limit (15) can be understood as a generalisation of Giventalâ&#x20AC;&#x2122;s mirror symmetry result [16]. In this case the gauged potential plays the roĚ&#x201A;le of the I function and the graph potential of the J function, whit the quantum Kirwan map as the mirror transformation. To obtain the actual I, J functions, we need to localise the potentials, as follows. Consider the case C = P equipped with the standard CĂ&#x2014; -action given by rotation, with fixed points 0, â&#x2C6;&#x17E; â&#x2C6;&#x2C6; P. Let denote the equivariant parameter. The graph potential Ď&#x201E;X T extends to a CĂ&#x2014; -equivariant potential Ď&#x201E;X T,CĂ&#x2014; : QH(X G) â&#x2020;&#x2019; Î&#x203A;X . The localised graph potential Ď&#x201E;X T,â&#x2C6;&#x2019; : QH(X T ) â&#x2020;&#x2019; QH(X T ) â&#x2C6;&#x2019;1 is the localisation to the fixed point 0 â&#x2C6;&#x2C6; P. Ď&#x201E;X T,â&#x2C6;&#x2019; is a solution to the fundamental quantum differential equation â&#x2C6;&#x201A;Ď&#x2026; Ď&#x201E;X T,â&#x2C6;&#x2019; (Îą) = Ď&#x2026; â&#x2C6;&#x2014;Îą Ď&#x201E;X T,â&#x2C6;&#x2019; (Îą) for the Frobenius manifold associated to the GW theory of X T . Localising the gauged T as well, and by considering a CĂ&#x2014; -equivariant extension potential Ď&#x201E;X,â&#x2C6;&#x2019; of the quantum Kirwan map ÎşTX , we have the localised adiabatic limit theorem (18)

T Ď&#x201E;X T,â&#x2C6;&#x2019; â&#x2014;Ś ÎşTX = lim ÎşT,class â&#x2014;Ś Ď&#x201E;X,â&#x2C6;&#x2019; , X tâ&#x2020;&#x2019;â&#x2C6;&#x17E;

where ÎşT,class is the classical Kirwan map. In the toric case X = X CN , T = (CĂ&#x2014; )k , the right term agrees with Giventalâ&#x20AC;&#x2122;s I function,

Applications of Gauged Gromov-Witten Theory: A Survey

(19)

κT,class X

◦

T τX,− (α)

= exp(α/ )

d∈H2T (X)

j=1

m=−∞ (µj

+ m )

m=−∞ (µj

+ m )

µj (d)

and the localised graph potential τX T,− is the J function, by Givental [15]. This implies that the quantum Kirwan map plays the rôle of the mirror map. For more details see [45, Section 8].

4.3

Wall-crossing for GW invariants of quotients.

We will follow the same notations we had in Section 2.1. We now explain how the quantised version of Kalkman’s wall crossing Equation (3), holds in quantum cohomology. Our wall crossing formula for Gauge GW invariants and the quantum Kirwan map descends to one for GW invariants of X G. We may think of gauged maps as interpolating between maps to the quotient and quotient of maps, thus taking both of the polarisations to the adiabatic large limit, and applying the factorisation of Equation (15) we obtain. Theorem 4.3.1 (Theorem 3.5 [21]). Applying the wall-crossing formula for two polarisations Lt+ and Lt− and taking the limit t → ∞ we have (20) (Wall-Crossing for GW potentials) τX − G ◦ κG X,− (α) =

τX + G ◦ κG X,+ (α)−

Res τX,ζ,Lδ (α).

In general the terms on the right of Equation (20) are gauged Gromov-Witten invariants, which in principle can be expressed in terms of the usual Gromov-Witten invariants of the smaller quotients on the walls, just as in classical Kalkman’s formula. However gauged invariants are easier to compute, so we will leave the formula as is, since it is easier to understand the effect of crossing a wall for the crepant case to be discussed in Section 4.4. Remark 4.3.2. The formulas just presented are in their most basic form. In general one can add insertions from classes β ∈ H(Mn (C)) as well as a twisting by Euler classes of index bundles. This is in particular important for explicit computations of particular invariants. We end this discussion with an example.

Eduardo GonzaĚ lez

Example 4.3.3. We use the same notation as in Example 3.1.6. Let G = CĂ&#x2014; acting on X = CN by scalar multiplication, HG (X) = Q[Îž]. And LÂą are the lines with weights Âą1, inducing X â&#x2C6;&#x2019; G = â&#x2C6;&#x2026; and X + G = PN â&#x2C6;&#x2019;1 . The wall is t = 0 corresponding to the point 0 â&#x2C6;&#x2C6; X. We compute genus zero, degree one three-point invariants ha , hb , hc 0,1 of Pkâ&#x2C6;&#x2019;1 via wall-crossing. Since the equivariant chern class cG 1 (X) = N Îž, the minimal Chern number of X is N , and this implies that the Kirwan map i i D0 ÎşG X (Îž ) = h , i â&#x2030;¤ N â&#x2C6;&#x2019; 1.

By dimension reasons, there are no quantum Kirwan corrections, and thus the quantum Kirwan map agrees with the classical one. The adiabatic limit (15) with insertions from the Fulton-McPherson space implies that the three-point invariants in the quotient equal the gauged Gromov-Witten invariant, a b c h ,h ,h = evâ&#x2C6;&#x2014;1 Îž a â&#x2C6;Ş evâ&#x2C6;&#x2014;2 Îž b â&#x2C6;Ş evâ&#x2C6;&#x2014;3 Îž c â&#x2C6;Ş f â&#x2C6;&#x2014; Î˛ G 0,1

[M3 (P,X,d)]

where Î˛ â&#x2C6;&#x2C6; H(M3 (P)) is the class fixing the location of the three marked points. Consider the wall-crossing Formula (20). There are no holoG morphic spheres in X, so the moduli stack M0 Îś (P, X, LÂą , Îś, t, d) is a point, consisting of the bundle P with first Chern class c1 (P ) = d â&#x2C6;&#x2C6; 2 (X, Z) â&#x2C6;ź Z (d = 1) with constant section equal to zero. Thus HG = GÎś M3 (P, X, LÂą , Îś, t, d) â&#x2C6;ź = M3 (P). By wall-crossing evâ&#x2C6;&#x2014;1 Îž a â&#x2C6;Ş evâ&#x2C6;&#x2014;2 Îž b â&#x2C6;Ş evâ&#x2C6;&#x2014;3 Îž c â&#x2C6;Ş f â&#x2C6;&#x2014; Î˛ ha , hb , hc = ResÎž Eul(Ind(T (X/G))+ ) 0,1 [M3 (P,Lâ&#x2C6;&#x2019; ,L+ ,X,Îś,t,1)] = ResÎž Îž a+b+c /Îž 2k 1 a + b + c = 2N â&#x2C6;&#x2019; 1 = 0 otherwise

which counts the unique line passing through two generic points and a generic hyperplane.

4.4

A GIT version of the crepant conjecture.

We now explain how we can use the wall-crossing formula to prove a special case of the crepant conjecture. A birrational transformation of GIT type induced by a wall-crossing L+ , Lâ&#x2C6;&#x2019; Ď&#x2020; : X â&#x2C6;&#x2019; G X + G

Applications of Gauged Gromov-Witten Theory: A Survey

will be called crepant if cC 1 (N F ) = 0 for each fixed component F ⊂ × M C of the master space, and weakly crepant if the sum of the weights of C× on N F , counted with multiplicity, vanishes. Recall that the fixed point sets of the master space are in correspondence with fixed sets X ζ by one-parameter subgroups as discussed in Section 2.1. The contribution from a fixed point component X ζ,t to the Formula (20) is given by the fixed point potential τX,ζ,t : QHG (X) → ΛG [ξ, ξ −1 ], which in turn is given by twisted gauged invariants coming from the G fixed moduli Mn (ζ, t, d) := Mn ζ (C, X, L± , ζ, t, d), where we reduce the notation for simplicity. Mn (ζ, t, d) consists of tuples (P, Ĉ, u) where P → C is a G-bundle and u : Ĉ → P (X) is ζ-fixed, in particular, the restriction of u to the principal component of C maps into the locus X ζ . Let C× ζ ⊂ Gζ denote the subgroup of Gζ generated by ζ ∈ gζ and identify ∼ Pic(C) = Maps(C, BC× ζ )=Z with the group of isomorphism classes of C× ζ -bundles Pζ → C. There is a canonical action of the Picard group Pic(C) on the moduli stack Mn (ζ, t, d), given by Pζ (P, Ĉ, u) = (P ×C× Pζ , Ĉ, u) ζ

where we use that (P ×C× Pζ )(X ζ ) ∼ = (P ⊗ OC (d))(X ζ ) ζ

ζ since the action of C× ζ on X is trivial. When restricting to the part induces an isomorphism shifting degree

(21)

Sδ : Mn (ζ, t, d) → Mn (ζ, t, d + δ)

where δ = c1 (Pζ ) is the generator. The action of Pic(C) lifts to the universal curves, and, since the obstruction theory on Mn (ζ, t, d) is the ζ-invariant part of Ind(T (X/G)), the isomorphism preserves the relative obstruction theories and so the Behrend-Fantechi virtual fundamental classes. Since the evaluation map is unchanged, the class ev∗ α is preserved for any α ∈ HGζ (X)n . The action of Pic(C) helps us understand the fixed point contributions τX,ζ,d,t . The contribution of any component Mn (ζ, t, d) of class d ∈ H2G (X) differs from that from the component induced by acting by Pζ , of class d + δ, by the difference in Euler classes Eul(Rp∗ ev∗ T (X/G)+ ) and Eul(Sδ∗ Rp∗ ev∗ T (X/G)+ ). We now compute this difference. Assume for simplicity that the component X ζ,t of the fixed point set X ζ

Eduardo González

which is semistable for t is connected (repeat the following argument for each connected component). Consider the decomposition into C× ζ bundles k νX ζ,t νX ζ,t = i

where k = codim(X ζ,t ) and C× ζ acts on νXiζ,t with non-zero weight µi ∈ ∗ Z. Then ev T (X/Gζ ) is isomorphic to Sδ ev∗ T (X/G) on the bubble components, since the Gζ -bundles are trivial, while on the principal component e∗ νX ζ,t and (e∗ T (X/G))+ ∼ = Sδ∗ (e∗ T (X/G))+ ∼ =

e∗ νX ζ,t ⊗ ev∗C OC (µi δ) i

where OC (1) is a hyperplane bundle on C. Hence the difference between the pull-back complexes vanishes on the bubble components and so pushes forward to a complex on the principal part of the universal curve p0 : C × Mn (ζ, t, d) → Mn (ζ, t, d). This is a representable morphism of stacks, which admit a presentation as global quotients embedded in smooth DM stacks [1, Theorem 1.0.2] in which case by Grothendieck-Riemann-Roch [14, Theorem 3.1] , (z ∗ e∗ νX ζ,t )⊕µi δ (22) Sδ∗ Ind(T (X/G))+ ∼ = Ind(T (X/G))+ ⊕ i

where z : Mn (ζ, t, d) → C × Mn (ζ, t, d) is a constant section of p0 . By the splitting principle we may assume that the νX ζ,t are line bundles. i The difference in Euler classes Eul(Sδ∗ Ind(T (X/G))+ ) Eul((Ind(T (X/G))+ )−1 ∈ HC× (Mn (ζ, t, d)) is given by the Euler class of the last summand in (22) k k ∗ ∗ ⊕µi δ Eul (z e νX ζ,t ) (µi ξ + c1 (νX ζ,t ))µi δ = = i

i=1

k i=1

i=1

(ξ + c1 (νX ζ,t )/µi )µi δ i

k i=1

µiµi δ

Applications of Gauged Gromov-Witten Theory: A Survey

factoring out the weights Âµi . Let Âµ = product we obtain (23)

i=1 Âµi .

(Î¾ + c1 (Î½X Î¶,t )/Âµi )Âµi Î´ = Î¾ Î´Âµ + Î¾ Î´Âµâ&#x2C6;&#x2019;1 i

i=1

ï£«

Î¾ Î´Âµâ&#x2C6;&#x2019;2 ï£Î´ 2

i =j

c1 (Î½X Î¶,t )c1 (Î½X Î¶,t ) + i

k i=1

Î´

Expanding out the

c1 (Î½X Î¶,t ) + i

i=1

ï£¶

Î´(Î´ â&#x2C6;&#x2019; 1/Âµi )c1 (Î½X Î¶,t )2 ï£¸ + . . . i

and . . . indicates further terms that are polynomials in Î¾, Î¾ â&#x2C6;&#x2019;1 , Î´, Âµi and crepant, the sum of the weights in the normal Âµâ&#x2C6;&#x2019;1 i . If the wall crossing is directions vanish, so Âµ = ki=1 Âµi = 0, and in (23), the exponents Î´Âµ vanish. Adding all expressions (23) over Î´, we obtain the wall crossing term (24) ResidÎ¾ q d+Î´ Ï&#x201E;X,Î¶,d+Î´,t = Î´â&#x2C6;&#x2C6;Z

ÂµiÂµi Î´ q d+Î´

Î´â&#x2C6;&#x2C6;Z,tâ&#x2C6;&#x2C6;(â&#x2C6;&#x2019;1,1) i=1

(1 + Î¾ â&#x2C6;&#x2019;1 [Mn (Î¶,d,t)]

Î´c1 (Î½X Î¶,t ) + . . .) evâ&#x2C6;&#x2014; Î±.

i=1

Now for any integers mi â&#x2030;¥ 0, ni , i = 1, . . . , k k Î´

Î´ mi ÂµiÂµi Î´â&#x2C6;&#x2019;ni q d+Î´ =a.e. 0

i=1

vanishes almost everywhere in q, being a function timesn a sum of derivatives of delta functions in q. (Here recall that nâ&#x2C6;&#x2C6;Z q = Î´(q = 1), the delta distribution supported at q = 1.) Almost everywhere equality in a formal parameter q means the following: after consider both sides as elements in Hom(H2G (X, Z)/ torsion, Q) the space of distributions on 2 (X, R)/H 2 (X, Z), the difference is tempered in at least one direcHG G tion and its Fourier transform in that direction has support of measure zero. Since Ï&#x201E;X + G â&#x2C6;&#x2019; Ï&#x201E;X â&#x2C6;&#x2019; G is a sum of such wall-crossing terms, the discussion above justifies the following result. Theorem 4.4.1 (Theorem 1.13 [21]). If the all the wall-crossings are weakly crepant then G Ï&#x201E;X â&#x2C6;&#x2019; G â&#x2014;¦ ÎºG X,â&#x2C6;&#x2019; =a.e Ï&#x201E;X + G â&#x2014;¦ ÎºX,+

almost everywhere (a.e.) in the formal parameters q.

Eduardo González

Note that this result is not a quantisation of any classic result in equivariant cohomology and it is purely a quantum cohomology result. Theorem 4.4.1 is a version of the crepant resolution conjectures of LiRuan [32], Bryan-Graber [6], Coates-Ruan [10]. It is important to remark that for derived categories (B-model) the work [24, 25], brings equivalences for derived categories of sheaves in the GIT case where the same crepant condition of weights is found.

4.5

Quantum Witten localisation and quantum abelianisation

We now discuss a quantum version of the abelianisation formula relating traces of the quotients X G, X T , as discussed in section 2.2. We first need to introduce a quantum version of Witten’s non-abelian localisation. Theorem 4.5.1 (Theorem 1.0.2 [19]). Under suitable stable=semistable conditions, the following formula holds (25) G (Quantum Witten localisation) τX = τX G ◦κG + τX,G,ζ,Lt . X [ζ],t∈(0,∞)

Theorem 4.5.1 follows from the wall crossing formula (11) and by taking the large (15), and small (14) limits simultaneously. The contributions on the right correspond to contributions described in (11) from all components fixed by one-parameter subgroups for t ∈ (0, ∞). Here we are assuming stable=semistable with respect to the smaller groups on each of the fixed moduli that contributes. As mention above, our original intention for Equation (25) was to show abelianisation for quantum cohomology. Recall from Section 2.2 the notations and discussion on Martin’s formula (5). Let T ⊂ G be a maximal torus, and suppose stable=semistable for the actions of T, G on the projective variety X. Let W denote the Weyl of T and let RestrG T : HG (X) → HT (X), be the natural restriction. Then, integration over X G and X T are related by −1 T G τX G ◦ κG X = |W | τX T ◦ κX ◦ RestrT . The abelianisation result states that in quantum cohomology a similar equation holds, after we change the traces by quantised potentials. Theorem 4.5.2 (Theorem 1.0.3 [19]). Under suitable stable=semistable

Applications of Gauged Gromov-Witten Theory: A Survey

conditions, (26)

T G −1 G τX G ◦ κG X = |W | πT ◦ τX T ◦ κX ◦ RestrT .

Recall that the potential τX T is twisted by the Euler class of the index bundle Index(g/t) induced by the roots, the κ’s denote quantum Kirwan maps. The push-forward πTG : H2T (X) → H2G (X) induces one of equivariant Novikov variables ΛTX → ΛG X ( see (9)) by G cd q d → cd q πT (d) . πTG : ΛTX → ΛG X; d∈H2T (X)

d∈H2T (X)

The map RestrG T : HG (X) → HT (X) induces the isomorphism W and together with the map π G in Novikov variables H HG (X)∼ (X) = T T we get, the map QHG (X) → QHT (X), by extending to quantum cohomology. The proof of the theorem is iterative. First we prove it in the small limit [22, Theorem 1.3], that is G T τX = |W |−1 πTG ◦ τX .

In the small chamber, abelianisation is somehow easier and similar in nature to the original proof given by Martin (Section 2.2) since the moduli spaces are essentially “GIT quotients” of stacks in the sense of Equation (12). There is a technical detail in this proof, namely, we chow (X) ⊂ QH (X), the portion of classes needed to restrict to QHG G which are algebraic. This is because we rely on Grothendieck-RiemannRoch for sheaf cohomology on stacks. (Although we believe that this restriction is not necessary, see [22, Section 5] for more details.) Once the proof of abelianisation in the small limit is established, the proof of Theorem follows by iteration, increasing the scaling t to pass to the big limit while crossing all walls in between. We then check that the contributions for both quotients arising on the walls agree. To be more precise, we carefully compare the wall terms in the abelian and nonabelian quantum Witten localisation formulas T G τX −τX G ◦κX,G = τX,G,ζ,t and τX −τX T ◦κX,T = τX,T,ζ,t [ζ] =0,t

[ζ] =0,t

Again, we remark that all the abelian potentials are twisted by the Euler class of the index bundle Ind(νg/t ). This now gives the abelianisation formula in the large limit, that is, for the GIT quotients.

Eduardo González

Example 4.5.3. We consider the Grassmann variety Gr(k, n) of kplanes in the affine space Cn of dimension n. We present Gr(k, n) in the standard GIT manner as follows. Let GLk act on X = Hom(Ck , Cn ) by (g · x)(v) = x(g −1 v) for all x ∈ X, v ∈ Ck and g ∈ G. Since 2 (X, Q)∼Q, we take the linearisation corresponding to cG (X). The HG = 1 semistable locus X ss is given by the injective maps (full rank matrices) in X. Then Gr(k, N ) = X GLk . Consider the maximal torus T = (C× )k of invertible diagonal matrices in G. The Weyl group W = Sn acts on the equivariant cohomology HT (X; Q) = Q[θ1 , . . . , θk ] and the invariant part HT (X; Q)W is identified with HG (X; Q). The roots are then identified with θi − θj for i < j, and the first Chern class cG 1 (X) = n(θ1 + · · · + θk ), thus it is positive, meaning that the equivariant Chern class takes postive values on all degrees for which there is a gauged map. Positivity (monotonity) of X 2 (X; Q) in implies that there are no quantum corrections coming from HG G κX (see Remark 8.7(b) in [45], or Remark 4.1.1 in [19]). Let ti denote the coordinates in HT (X; Q), then by the localised gauged potential associated to the torus (19) is τX,T,− (t0 + t1 θ1 + · · · + tk θk ) = exp(t0 + t1 θ1 + · · · + tk θk )

d∈H2T (X;Q)

exp(

ti di )τX,T,− (d)q d ,

where the degree d = d1 θ1 + · · · + dk θk component is given by

τX,T,− (d) =

di −dj

m=−∞ ((θi 0 m=−∞ ((θi i =j,i,j≤k

k − θj ) + m )

− θj ) + m ) j=1

m=−∞ (θj

+ m )n

m=−∞ (θj

+ m )n

whose first term is the contributions from the roots and the second is ntimes repeated. By the abelianisation formula (18), the localised graph potentials on H 2 (Gr(k, n)) is given by (27)

T G τX G,− = πTG ◦ κclass X,T ◦ τX,− ◦ RestrT ,

since the quantum Kirwan map has no quantum correction. This is the original formula conjectured by Hori-Vafa [26, Appendix A] and proved by Bertram et al [3, Theorem 1.5] which led to the general Abelian-nonAbelian correspondence [4, Conjectures 1.1, 1.2].

Applications of Gauged Gromov-Witten Theory: A Survey

Example 4.5.4. We extend the example above to show that similar computations hold even if the case when the Kirwan map has corrections. In this case however, the corrections can be hard to compute, as expected since they are related to the mirror map. Let V denote a left GLk -module. We denote by VGr(k,n) → Gr(k, n) the associated vector bundle (X ss × V )/GLk , where the action is diagonal. Let 0 → U → OGr(k,n) ⊗ CN → Q → 0 denote the tautological sequence associated to the universal k-plane bundle U → Gr(k, n), where Q is the universal quotient. If W is the standard representation of GLk , then WGr(k,n) = U . For our example, consider l ≤ k, and the projectivisation F = P(Λl U ) → Gr(k, n) which has a presentation as a GIT quotient (X × Λl W ) (GLk × C× )

where C× acts on the fibres diagonally. The same computation above carries ad-verbatim by noticing that there is an extra generator HT (X × Λl W ) = Q[θ1 , . . . , θk , θk+1 ], and that the group GLk × C× has the same roots as in Example 4.5.3. × ∨ Notice k that T × C acts onl W with weights −θi , i = 1,∨. . . , k so on the l dimensional space Λ W with sums of l distinct −θi . Then the associated abelian localised potential twisted by the root contributions is given by essentially the same formula above, by adding tk+1 , θk+1 terms appropriately and for d = d1 θ1 + · · · + dk θk + dk+1 θk+1 , (28) τX×Λl W,T ×C× ,− (d) =

di −dj

m=−∞ ((θi

− θj ) + m )

− θj ) + m ) 0 k 0 n (kl) m=−∞ (θj + m ) m=−∞ (θk+1 + m ) , dj Dk+1 (d) n (kl) (θ + m ) k+1 j=1 m=−∞ (θj + m ) m=−∞

i =j,i,j≤k

m=−∞ ((θi

where Dk+1 (d) is dk+1 − d1 − · · · − dk . The fundamental solution is given by the localised graph potential (18) and (26). Here there might be corrections in the Kirwan map, for instance in the semi-positive case, when n = kl . As a more concrete example, consider the Grassmanian

Eduardo González

G(4, 6) and take F = P(Λ2 U ∨ ). The acts on X × Λ2 W with weight matrix  0 0  0 0  0 0 A=   0 0 0 0 0 0

maximal torus T = (C× )4 × C× 0 − 0 − 0 − − 0

     

Here the represents the vector = (1 1 1 1 1 1). Recall that (λ1 , λ2 , λ3 , λ4 ) ∈ (C× )4 acts on (4, 6) matrices by multiplying the ith column by λi , with weight 1. Thus represent the weights of the action on the entries in each column. (C× )4 acts on the six dimensional space Λ2 W with −1 on each term, these are the − in the last column. The last row represents the weights of C× acting on the fibres. Since the space is semi-positive, we need to account the corrections in the Kirwan (mirror) map, and thus the fundamental solution is computed by (26). Acknowledgement The author would like to thank Ernesto Lupercio, Sergei Galkin, Leticia Brambila-Paz and Ugo Bruzzo, for illuminating conversations. The author also thanks the department of mathematics at the Massachusetts Institute of Technology for their hospitality during the revisions of this note. Eduardo González Department of Mathematics, University of Massachusetts Boston, 100 William T. Morrissey Boulevard, Boston, MA 02125 eduardo@math.umb.edu

References [1] Dan Abramovich, Tom Graber, Martin Olsson, and Hsian-Hua Tseng. On the global quotient structure of the space of twisted stable maps to a quotient stack. J. Algebraic Geom., 16(4):731–751, 2007. [2] V. Balaji. Lectures on principal bundles. In Moduli spaces and vector bundles, volume 359 of London Math. Soc. Lecture Note Ser., pages 2–28. Cambridge Univ. Press, Cambridge, 2009. [3] Aaron Bertram, Ionuţ Ciocan-Fontanine, and Bumsig Kim. Two proofs of a conjecture of Hori and Vafa. Duke Mathematical Journal, 126(1):101–136, 2005.

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[4] Aaron Bertram, Ionuţ Ciocan-Fontanine, and Bumsig Kim. Gromov-witten invariants for abelian and nonabelian quotients. Journal of Algebraic Geometry, 17(2):275–294, 2008. [5] K. Behrend and B. Fantechi. 128(1):45–88, 1997.

The intrinsic normal cone.

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[6] Jim Bryan and Tom Graber. The crepant resolution conjecture. In Algebraic geometry—Seattle 2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 23–42. Amer. Math. Soc., Providence, RI, 2009. [7] M. Brion and C. Procesi. Action d’un tore dans une variété projective. In A. Connes et al., editors, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, volume 62 of Progress in Mathematics, pages 509–539, Paris, 1989, 1990. Birkhäuser, Boston. [8] Kai Cieliebak, A. Rita Gaio, Ignasi Mundet i Riera, and Dietmar A. Salamon. The symplectic vortex equations and invariants of Hamiltonian group actions. J. Symplectic Geom., 1(3):543–645, 2002. [9] Kai Cieliebak, Ana Rita Gaio, and Dietmar A. Salamon. J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions. Internat. Math. Res. Notices, (16):831–882, 2000. [10] Tom Coates and Yongbin Ruan. Quantum cohomology and crepant resolutions: A conjecture. arXiv 0710.5901. [11] Kai Cieliebak and Dietmar Salamon. Wall crossing for symplectic vortices and quantum cohomology. Math. Ann., 335(1):133–192, 2006. [12] Igor V. Dolgachev and Yi Hu. Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math., (87):5–56, 1998. With an appendix by Nicolas Ressayre. [13] D. Edidin. Equivariant algebraic geometry and the cohomology of the moduli space of curves. In Handbook of Moduli, Vol. I, Adv. Lect. Math. (24): 259–292, 2013. [14] Dan Edidin. Riemann-Roch for Deligne-Mumford stacks. A celebration of algebraic geometry, Clay Math. Proc., (18): 241–266, 2013. [15] A. B. Givental. Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices, (13):613–663, 1996. [16] Alexander Givental. A mirror theorem for toric complete intersections. In Topological field theory, primitive forms and related topics (Kyoto, 1996), volume 160 of Progr. Math., pages 141–175. Birkhäuser Boston, Boston, MA, 1998. [17] V. Guillemin and S. Sternberg. Birational equivalence in the symplectic category. Invent. Math., 97(3):485–522, 1989. [18] Ana Rita Pires Gaio and Dietmar A. Salamon. Gromov-Witten invariants of symplectic quotients and adiabatic limits. J. Symplectic Geom., 3(1):55–159, 2005. [19] Eduardo Gonzalez and Chris Woodward. Quantum Witten localization and abelianization of qde solutions. arXiv 0811.3358.

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[20] Eduardo Gonzalez and Chris Woodward. Quantum cohomology and toric minimal model programs. arXiv 1207.3253. [21] Eduardo Gonzalez and Chris T. Woodward. A wall-crossing formula for Gromov-Witten invariants under variation of GIT quotient. arXiv 1208.1727. [22] Eduardo Gonzalez and Chris Woodward. Gauged Gromov-Witten theory for small spheres. Math. Z., 273(1-2):485–514, 2013. [23] Tom Graber and Rahul Pandharipande. Localization of virtual classes. Invent. Math., 135(2):487–518, 1999. [24] Daniel Halpern-Leistner. The derived category of a GIT quotient. arXiv 1203.0276. [25] Daniel Halpern-Leistner and Ian Shipman. Autoequivalences of derived categories via geometric invariant theory. arXiv, 1303.5531. [26] Kentaro Hori and Cumrun Vafa. Mirror symmetry. arXivhep-th/0002222. [27] J. Kalkman. Cohomology rings of symplectic quotients. J. Reine Angew. Math., 485:37–52, 1995. [28] F. C. Kirwan. Cohomology of Quotients in Symplectic and Algebraic Geometry, volume 31 of Mathematical Notes. Princeton Univ. Press, Princeton, 1984. [29] Young-Hoon Kiem and Jun Li. A wall crossing formula of donaldson-thomas invariants without chern-simons functional. arXiv 0905.4770v2. [30] M. Kontsevich and Yu. Manin. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys., 164(3):525–562, 1994. [31] G. Kempf and L. Ness. The length of vectors in representation spaces. In K. Lønsted, editor, Algebraic Geometry, volume 732 of Lecture Notes in Mathematics, pages 233–244, Copenhagen, 1978, 1979. Springer-Verlag, BerlinHeidelberg-New York. [32] An-Min Li and Yongbin Ruan. Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Invent. Math., 145(1):151–218, 2001. [33] S. Martin. Symplectic quotients by a nonabelian group and by its maximal torus. arXiv math.SG/0001002. [34] D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge. Springer-Verlag, Berlin-Heidelberg-New York, third edition, 1994. [35] Ignasi Mundet i Riera. Yang-Mills-Higgs theory for symplectic fibrations. PhD thesis, Universidad Autónoma de Madrid, 1999. [36] Ignasi Mundet i Riera. A Hitchin-Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math., 528:41–80, 2000. [37] P.-E. Paradan. The moment map and equivariant cohomology with generalized coefficients. Topology, 39(2):401–444, 2000. [38] P.-E. Paradan. Localization of the Riemann-Roch character. J. Funct. Anal., 187(2):442–509, 2001. [39] A. Ramanathan. Moduli for principal bundles over algebraic curves. I. Proc. Indian Acad. Sci. Math. Sci., 106(3):301–328, 1996.

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[40] A. Ramanathan. Moduli for principal bundles over algebraic curves. II. Proc. Indian Acad. Sci. Math. Sci., 106(4):421–449, 1996. [41] Michael Thaddeus. Geometric invariant theory and flips. J. Amer. Math. Soc., 9(3):691–723, 1996. [42] Edward Witten. Two-dimensional gauge theories revisited. J. Geom. Phys., 9:303–368, 1992. [43] Edward Witten. Phases of N = 2 theories in two dimensions. In Mirror symmetry, II, volume 1 of AMS/IP Stud. Adv. Math., pages 143–211. Amer. Math. Soc., Providence, RI, 1997. [44] Chris T. Woodward. Localization via the norm-square of the moment map and the two-dimensional Yang-Mills integral. J. Symp. Geom., 3(1):17–55, 1996. [45] Chris T. Woodward. Quantum Kirwan morphism and Gromov-Witten invariants of quotients I. To appear in Transform. Groups, 2015, arXiv 1204.1765. [46] Chris T. Woodward. Quantum Kirwan morphism and Gromov-Witten invariants of quotients II. To appear in Transform. Groups, 2015, arXiv 1408.5864. [47] Chris T. Woodward. Quantum Kirwan morphism and Gromov-Witten invariants of quotients III. To appear in Transform. Groups, 2015, arXiv 1408.5869. [48] Chris T. Woodward and Fabian Ziltener. Functoriality for Gromov-Witten invariants under symplectic quotients. [49] F. Ziltener. Symplectic vortices on the complex plane and quantum cohomology. PhD thesis, Zurich, 2006.

Morfismos, Vol. 22, No. 2, 2018, pp. 53–83 Morfismos, Vol. 22, No. 2, 2018, pp. 53–83

Derived Mackey functors and profunctors: an overview of results Derived Mackey functors and profunctors: an overview D. Kaledinof1 results D. Kaledin

Abstract In this paper we overview the theory of derived Mackey functors and profunctors. Abstract In this paper we overview the theory of derived Mackey functors 2010 Mathematics Subject Classification: 14A22 and profunctors.

Keywords and phrases: Mackey functors, profuctors. 2010 Mathematics Subject Classification: 14A22 Keywords and phrases: Mackey functors, profuctors.

Introduction

“Mackey functors” associated to a finite group G have long been a stan1 Introduction dard tool both in group theory and in algebraic topology (specifically, in the G-equivariant homotopy theory). notion was origi“Mackey functors”stable associated to a finite groupThe G have long been a stannallydard introduced byinDress and later clarified by several in tool both group[2]theory and in algebraic topologypeople, (specifically, particular Lindner [7].stable The reader can theory). find modern in the by G-equivariant homotopy The expositions notion was inorigithe topological contextbye.g. in [6], or aclarified more algebraic treatment nally introduced Dress [2] [8], and[1], later by several people, in in [9]. particular by Lindner [7]. The reader can find modern expositions in For finite group G, G-Mackey form an abelian category the any topological context e.g. in [6], functors [8], [1], or a more algebraic treatment M(G). If one wants to develop a homological theory of Mackey functors, in [9]. it seemsFor natural to consider theG-Mackey derived category However, any finite group G, functors D(M(G)). form an abelian category thereM(G). is an alternative suggested in [4]. By modifying the functors, very If one wantstotothis develop a homological theory of Mackey definition of anatural Mackeytofunctor, onethe canderived construct a triangulated category it seems consider category D(M(G)). However, DM(G) Mackey that contains M(G) but differs thereofis “derived an alternative tofunctors” this suggested in [4]. By modifying the very fromdefinition D(M(G)). category DM(G) reflects better the properties of of aThe Mackey functor, one can construct a triangulated category the G-equivariant stable homotopy category,that andcontains it also behaves better DM(G) of “derived Mackey functors” M(G) but differs thanfrom D(M(G)) from The the purely formal pointreflects of view. For more details, of D(M(G)). category DM(G) better the properties we refer the reader to [4]; here we only mention the G-equivariant stable homotopy category,two andthings: it also behaves better 1 than D(M(G)) from the purely formal point of view. For more details, Partially supported by RScF, grant number 14-21-00053, for AG Laboratory we refer the toFoundation [4]; here we only mention two things: SU-HSE, and by thereader Dynasty award. 1 Partially supported by RScF, grant number 14-21-00053, for AG Laboratory SU-HSE, and by the Dynasty Foundation 53 award.

D. Kaledin

• DM(G) has a natural semiorthogonal decomposition into pieces that are identified with the derived categories of representations of certain subquotients of G; the gluing functors of this descomposition have a natural interpretation in terms of a certain generalization of Tate cohomology of finite groups. • DM(G) has a lot of autoequivalences – in particular, any finitedimensional real representation V of G gives rise to a “suspension autoequivalence” ΣV : DM(G) → DM(G), a generalization of the homological shift. Another way to extend to the notion of a Mackey functor is to consider more general groups G. In fact, if one allows groups with non-trivial topology, then G does not have to be finite – one can consider any compact Lie group. However, from the point of view of algebra, a more natural modification is to allow groups that are discrete but infinite. Formally, one can do so without any changes – Lindner’s definition works for any group G, albeit the resulting category only depends on its profi However, once more there is an alternative suggested nite completion G. in [5] under the name of a “G-Mackey profunctor”. Mackey profunctors seem to reflect better the structure of the group G. In particular, in good situations, giving a G-Mackey profunctor M is equivalent to giving a system of Mackey functors MN ∈ M(G/N ), N ⊂ G a cofinite normal subgroup, related by some natural compatibility isomorphisms. One can also combine the two stories and develop the notion of a “derived Mackey profunctor”. This has also been done in [5]. It turns out that even if one is interested in non-derived Mackey profunctors, looking at them in the derived context is cleaner and gives stronger results. Unfortunately, both [4] and [5] are rather long and technical papers. The goal of the present paper is to give a brief overview of them. We do not give any proofs whatsoever, and we keep the technicalities to an absolute minimum. Instead, we try to present the conceptual ideas behind the constructions, and to illustrate the theory by some key examples. The paper is organized as follows. Section 1 is a recollection of the standard theory of Mackey functors. Section 2 is concerned with derived Mackey functors for a finite group G. Generally, we follow [4], but we use improvements suggested in [5] to clean up some statements and simplify the proofs. In particular, we describe and use a relation between Mackey functors and finite pointed G-sets. We illustrate the

Derived Mackey functors and profunctors: an overview of results

theory in the simplest possible example G = Z/pZ, the finite cyclic group of prime order. In Section 3, we turn to the theory of Mackey profunctors developed in [5]. The main example here is G = Z, the infinite cyclic group.

Notation. For any category E, we denote by E o the opposite category. For every integer n ≥ 0, we denote by [n] the set of integers i, 0 ≤ i ≤ n, and as usual, we denote by ∆ the category of the sets [n] and order-preserving maps g between them. Simplicial objects in a category E are functors from ∆o to E. For any small category I and ring R, we denote by Fun(I, R) the abelian category of functors from I to the category of R-modules, and we denote by D(I, R) the derived category of the abelian category Fun(I, R). For any functor γ : I → I between small categories, we have the natural pullback functor γ ∗ : Fun(I, R) → Fun(I , R), and it has a left and a right adjoint γ! , γ∗ : Fun(I , R) → Fun(I, R), the left and right Kan extension functors. The derived functors L γ! , R γ∗ : D(I , R) → D(I, R) are left and right-adjoint to the pullback functor γ ∗ : D(I, R) → D(I , R).

Recollection on Mackey functors.

We start with a brief recapitulation of the usual theory of Mackey functors (we follow the exposition in [5, Section 2]). Assume given a category C that has fibered products. The category QC is the category wwith same objects as C, and with morphisms from c1 to c2 given by isomorphism classes of diagrams (1)

c1 ←−−−− c −−−−→ c2

in C. The compositions are obtained by taking pullbacks. Note that we have a natural embedding (2)

e : C o → QC

that is identical on objects, and sends a map f to the diagram (1) with l = f and r = id. The category QC is obviously self-dual: QC o is identified with QC by the functor that flips r and l in (1). By duality, the embedding (2) induces an embedding eo : C → QC.

D. Kaledin

Now fix a group G and a ring R, and take as C the category ΓG of finite G-sets, that is, finite sets equipped with an action of the group G. The category ΓG has fibered products, so we can define the category QΓG and the embedding (2). Definition 2.1. An object E ∈ Fun(ΓoG , R) is additive if for any S1 , S2 ∈ ΓoG , the natural map (3)

E(S1 S2 ) → E(S1 ) ⊕ E(S2 )

is an isomorphism. An R-valued G-Mackey functor E is an object E ∈ Fun(QΓG , R) whose restriction e∗ E ∈ Fun(ΓoG , R) is additive. The full subcategory in Fun(QΓG , R) formed by Mackey functors is denoted by M(G, R). we denote by Funadd (ΓoG , R) ⊂ Fun(ΓoG , R) the full subcategory spanned by additive objects, and let OG ⊂ ΓG be the subcategory of G-orbits – that is, G-sets of the form [G/H], H ⊂ G a subgroup. Since every finite G-set decomposes into a disjoint union of G-orbits, o ⊂ Γo provides an equivalence (3) insures that restriction to OG G o Funadd (ΓoG , R) ∼ , R). = Fun(OG

In other words, an additive object E ∈ Fun(ΓoG , R) is uniquely determined by specifyings its values E H = E([G/H]) at all G-orbits, and maps f ∗ = E(f ) : E H → E H , one for each map f : [G/H ] → [G/H] of G-orbits, compatible with compositions for composable pairs of maps f , f . To extend E to a Mackey functor, one needs to also specify a map (4)

f∗ = E(eo (f )) : E(S ) → E(S)

for any map f : S → S of finite G-sets. These are sometimes called transfer maps (this is how they appear in equivariant stable homotopy theory). By additivity, it suffices to specify the transfer maps for maps G-orbits. They should be compatible with compositions, and for any two maps f : S → S, f : S → S, there is a certain compatibility condition between f∗ and f ∗ encoded in the definition of the category QΓG . Explicitly, if S = [G/H], S = [G/H ], S = [G/H ], and f , f

Derived Mackey functors and profunctors: an overview of results

are induced by embeddings H , H ⊂ H, the fibered product S ×S S decomposes into a disjoint union (5) S ×S S = [G/Hs ] s∈S

of G-orbits indexed by the finite set S = H \H/H , and we must have (6) f ∗ ◦ f∗ = ◦ f s∗ , f s∗ s∈S

: [G/Hs ] → : [G/Hs ] → [G/H ] are projections of where the component G/Hs of the decomposition (5). This is known as the double coset formula. For any subgroup H ⊂ G, we have a natural restriction functor H ρ : ΓG → ΓH sending a G-set S to the same set S considered as an H-set. The functor ρH has a left-adjoint γ H : ΓH → ΓG . Both ρH and γ H preserve fibered products, thus induce functors (7)

[G/H ],

Q(γ H ) : QΓH → QΓG ,

Q(ρH ) : QΓG → QΓH .

One checks that the adjunction between ρH and γ H induces an adjunction between Q(γ H ) and Q(ρH ), and that the functor (8)

ΨH = Q(ρH )! ∼ = Q(γ H )∗ : Fun(QΓG , R) → Fun(QΓH , R)

preserves additivity, thus sends Mackey functors to Mackey functors. The functor ΨH is known as the categorical fixed points functor. On the other hand, for any subgroup H ⊂ G with normalizer NH ⊂ G, the quotient W = NH /H = AutG ([G/H]) acts on the fixed points subset S H of any G-set S. Then sending S to S H gives a functor ϕH : ΓG → ΓW preserving fibered products, and we can consider the adjoint pair of functors (9)

ΦH = Q(ϕH )! : Fun(QΓG , R) → Fun(QΓW , R),

InflH = Q(ϕH )∗ : Fun(QΓW , R) → Fun(QΓG , R).

Both preserve additivity, thus induce adjoint functors between M(G, R) and M(W, R). These are the geometric fixed points functor and the inflation functor. The embedding M(G, R) ⊂ Fun(QΓG , R) admits a left-adjoint additivization functor Add. This allows to define tensor products of Mackey functors. Namely, the cartesian product functor (10)

m : ΓG × ΓG → ΓG

D. Kaledin

preserves fiber products, thus induces a functor Q(m) : QΓG × QΓG → QΓG . Then for any two algebras R1 , R2 over a commutative ring k and Mackey functors E1 ∈ M(G, R1 ), E2 ∈ M(G, R2 ), their tensor product E1 ◦ E2 is given by (11)

E1 ◦ E1 = Add(Q(m)! (E1 k E2 )) ∈ M(G, R1 ⊗k R2 ).

The tensor product (11) is unital, with the unit given by the so-called Burnside Mackey functor A ∈ M(G, Z). Explicitly, for any subgroup H ⊂ G, AH is canonically identified with the Burnside ring AH of the group H given by (12)

AH = Z[Iso(ΓH )]/{[S S ] − [S] − [S ]},

that is, the free abelian group generated by isomorphism classes [S] of finite H-sets S, modulo the relations [S S ] = [S] + [S ] (this is a commutative ring, with the product induced by the cartesian product of H-sets). Note that since any G-set uniquely decomposes into a disjoint union of G-orbits, we have (13) AG ∼ Z, = H⊂G

where the sum is over all conjugacy classes of subgroups H ⊂ G. Both the categorical fixed points functor ΨH and the geometric fixed points functor ΦH commute with the product (11). If R is a commutative ring, then the product (11) with R1 = R2 = k = R turns M(G, R) into a unital symmetric tensor category. An R-valued G-Green functor is an algebra object in the category M(G, R). For any subgroup H ⊂ G, the functor γ H and the categorical fixed points functor ΨH can be computed rather explicitly. The geometric fixed points functor ΦH is harder to describe. However, there is the following result. Lemma 2.2 ([5, Lemma 2.4]). Assume given a normal subgroup N ⊂ G, with the quotient W = G/N . Then the inflation functor InflN is fully faithful. Moreover, for any E ∈ M(G, R), the adjunction map (14)

M → InflN ΦN M

Derived Mackey functors and profunctors: an overview of results

is surjective, and for any S ∈ ΓW , we have a short exact sequence (15)

f :S →S

f∗

M (S ) −−−−→ M (S) −−−−→ ΦN (M )(S) −−−−→ 0,

where f∗ is as in (4), and the sum is over all maps f : S → S in ΓG such that S has no elements fixed under N ⊂ G. Note that by additivity of E ∈ Fun(QΓG , R), the image of the map f∗ in (15) is not only the sum of the images of individual maps f∗ , it is actually the union of these images.

Derived Mackey functors.

We now turn to the derived version of the story of Section 2. To see why the derived category D(M(G, R)) might be a wrong object to work with, it suffices to recall the definition of the category QC: we only consider the isomorphism classes of diagrams (1), and completely ignore the fact that those diagrams can have non-trivial automorphism groups with nontrivial homology. On the formal level, the problem appears for example in Lemma 2.2: if one considers the derived categories D(M(G, R)), D(M(W, R)), then the functor InflN : D(M(W, R)) → D(M(G, R)) induced by the inflation functor (9) is not longer fully faithful. To obtain the correct category of derived Mackey functors, one has to take account of the automorphisms of diagrams (1) and treat QΓG as what it naturally is – a 2-category rather than simply a category. But then, one has to make sense of the derived category D(QΓG , R).

3.1

Quotient constructions.

In fact, [4] introduces not one but two equivalent ways to do it. Firstly, one can observe that any small category C defines a small additive category B C with the same objects, and morphism groups given by B C (c, c ) = Z[C(c, c )], where C(c, c ) is the set of morphisms from c to c in the category C. Giving a functor from C to R-modules is equivalent to giving an additive

D. Kaledin

functor from B C to R-modules. If C is not a category but a 2-category, then C(c, c ) are not sets but small categories. What one can do is consider the geometric realizations |C(c, c )| of the nerves of the categories C(c, c ), and let B C = C (|C(c, c )|, Z)

be the singular chain complexes of these geometric realizations. If C is a strict 2-category – that is, compositions are associative on the nose – then geometric realization is sufficiently functorial so that B C (−, −) define a DG category B C with the same objects as C. In the general case, it is still possible to define B C as an A∞ -category. Then one can define D(C, R) as the dervied category of A∞ -functors from B C to complexes of R-modules. This is done in [4, Subsection 1.6]. The second approach is that of [4, Section 4]. It uses the nerves more directly. Namely, recall that the nerve N (C) : ∆o → Sets of a small category C is a simplicial set whose value at [n] ∈ ∆o is the set of all diagrams (16)

fn−1

c0 −−−−→ . . . −−−−→ cn

in C. For any map g : [n] → [n ] in ∆, a diagram c ∈ N (C)([n ]) induces a diagram g ∗ c ∈ N (C)([n]) such that g ∗ ci = cg(i) . One can turn N (C) into another small category N (C) by applying the Grothendieck construction of [3]: the objects of N (C) are pairs [n], c of an object [n] ∈ ∆ and a diagram (16), and morphisms from [n], c to [n ], c are morphisms g : [n] → [n ] in ∆ such that c = g ∗ c . Sending a diagram (16) to cn gives a functor q : N (C) → C, and the corresponding pullback functor q ∗ : D(C, R) → D(N (C), R) is a fully faithful embedding. To characterize its image, say that a morphism in N (C) defined by g : [n] → [n ] is special if g(n) = n , and say that E ∈ D(N (C), R) is special if the map E(f ) is invertible for any special f . Then D(C, R) ⊂ D(N (C), R) is exactly the full subcategory spanned by special objects. Now, if one starts with a 2-category C, then diagrams (16) form a category rather than a set. However, one can still define the category N (C): objects are pairs [n], c , morphisms from [n], c to [n ], c are pairs of a morphism g : [n] → [n ] and a morphism c → g ∗ c . Then one can keep the same notion of a special map and special object, and define D(C, R) as the full subcategory in D(N (C), R) spanned by special objects.

Derived Mackey functors and profunctors: an overview of results

As it turns out, the second approach is much better for practical applications: one does not need to keep track of all the higher A∞ operations, since they are all packed into the structure of the category N (C). For applications to Mackey functors, then, one would start with a category C that has fibered products, define a category QC with the same objects and morphisms given by isomorphisms classes of diagrams (17)

c ←−−−− c −−−−→ c

in C, and refine it to a 2-category QC by letting QC(c, c ) be the groupoids of such diagrams and invertible maps between them. Then one would consider the nerve N (QC). This is more-or-less what is done in [4, Section 4] and [5, Section 4], but with a small modification. It turns out that instead of N (QC), one can use a smaller category SC. Its objects are pairs [n], c of [n] ∈ ∆ and a diagram (16) in C, and morphisms from [n], c to [n ], c are pairs g, α of a morphism g : [n] → [n ] and a collection of morphisms αi : c g(i) → ci , 0 ≤ i ≤ n, such that for any i and j, 1 ≤ i ≤ j ≤ n , the diagram c g(i) −−−−→ c g(j)    αj αi ci

−−−−→ cj

is a fibered product square in the category C. A map g, α in SC is special if g(n) = n and αn is invertible, and an object E ∈ D(SC, R) is special if E(f ) is invertible for any special map f . One denotes by (18)

DS(C, R) ⊂ D(SC, R)

the full subcategory spanned by special objects. One advantage of the category SC is that one can define an analog of the embedding (2) without replacing the category C o with its nerve: by definition, sending c ∈ C to the pair [0], c gives a natural functor e : C o → SC. Thus every object E ∈ D(SC, R) defines an object (19)

E = e∗ E ∈ D(C o , R)

D. Kaledin

that we call the base part of E. Sending a diagram (16) to cn gives a natural functor q : SC → QC, and the composition q ◦ e is the natural embedding (2). For every special map f in SC, q(f ) is invertible in QC, so that we have a natural functor (20)

q ∗ : D(QC, R) → DS(C, R).

This functor is in general not an equivalence, and we take DS(C, R) as the correct definition of D(QC, R), a refinement of D(QC, R). Although q ∗ is not an equivalence, note that the standard t-structure on D(SC, R) induces a t-structure on DS(C, R), q ∗ is t-exact, and it identifies the heart of the t-structure on DS(C, R) with Fun(QC, R). For any E ∈ Fun(QC, R), the base part of q ∗ E is e∗ E ∈ Fun(C o , R); extending this base part to a special object in D(SC, R) is equivalent to providing the transfer maps (4). The category DS(C, R) of (18) is functorial in C in the following sense. Say that a functor F : C → C between two small categories with fibered product is a morphism if it preserves fibered products. Then any morphism F induced a functor S(F ) : SC → SC that commutes with projections to ∆ and sends special maps to special maps. Therefore we have a pullback functor S(F )∗ : DS(C , R) → DS(C, R). It has been shown in [5, Corollary 4.15] that S(F )∗ has a left-adjoint functor S(F )! : DS(C, R) → DS(C, R).

3.2

Definitions and properties.

Now return to the situation of Section 2: fix a finite group G, and let ΓG be the category of finite G-sets. This category has fiberd products, so that for any ring R, we can consider the derived category DS(ΓG , R) of (18). Definition 3.2.1. An object E ∈ D(ΓoG , R) is additive if the natural map (3) is an isomorphism for any S1 , S2 ∈ ΓG . A derived R-valued G-Mackey functor is an object E ∈ DS(ΓG , R) with additive base part (19). We denote the full subcategory of derived R-valued G-Mackey functors by DM(G, R) ⊂ DS(ΓG , R).

Derived Mackey functors and profunctors: an overview of results

The standard t-structure on DS(ΓG , R) induces a t-structure on DM(G, R) whose heart is identified with M(G, R) by the functor q ∗ of (20). It has been proved in [5, Lemma 8.3] that the embedding DM(G, R) ⊂ DS(ΓG , R) admits a left-adjoint additivization functor Add : DS(ΓG , R) → DM(G, R).

(21)

If G = {e} is the trivial group consisting of its unity element e, then DM(G, R) ∼ = D(R) is the derived category of the category of R-modules (note that this is a non-trivial statement that requires a proof, such as the one found in [4, Subsection 3.2]). As in (7), for any subgroup H ⊂ G, we have the adjoint pair of functors ρH , γ H ; both are morphisms and induce an adjoint pair of functors S(γ H ) : SΓH → SΓG ,

S(ρH ) : SΓG → SΓH .

The functor S(γ H )∗ ∼ = S(ρH )! : DS(ΓG , R) → DS(ΓH , R) sends additive objects to additive objects, thus induces a functor (22)

ΨH : DM(G, R) → DM(H, R).

This is the derived counterpart of the categorical fixed points functor (8). On the other hand, for any H ⊂ G with normalizer NH and quotient W = NH /H, the fixed points functor ϕH : ΓG → ΓW is also a morphism, and induces an adjoint pair of functors (23)

ΦH = S(ϕH )! : DS(ΓG , R) → DS(ΓW , R),

InflH = S(ϕH )∗ : DS(ΓW , R) → DS(ΓG , R),

a derived counterpart the geometric fixed points functor and the inflation functor of (9). As in the non-derived case, both ΦH and InflH preserve additivity, thus induce an adjoint pair of functors between DM(G, R) and DM(H, R) (for ΦH , this is [5, Lemma 6.14]). As in Lemma 2.2, for any normal subgroup N ⊂ G with quotient W = G/N , the inflation functor InflN : DM(W, R) → DM(G, R) is fully faithful (this is [5, Lemma 6.12]).

D. Kaledin

To define products of derived Mackey functors, one considers the cartesian product functor (10). It is a morphism, but S(m)! does not automatically preserve additivity. So, for any two algebra R1 , R2 over a commutative ring k and any E1 ∈ DM(G, R1 ), E2 ∈ DM(G, R2 ), we let (24)

E1 ◦ E2 = Add(S(m)! (E1 k E2 )) ∈ DM(G, R1 ⊗k R2 ),

where Add is the additivization functor (21). This product has exactly the same properties as the underived product (11). In particular, the fixed points functors ΨH , ΦH are tensor functors. Moreover, we have a derived version A ∈ DM(G, Z) of the Burnside Mackey functor that serves as the unit object for the product, and a derived Burnside ring AG = A ([G/G]). Explicitly, as shown in [5, Subsection 8.3], we have C (WH , Z), AG = H⊂G

where the sum is over all the conjugacy classes of subgroups H ⊂ G, and the corresponding summand is the homology complex of the group WH = AutG ([G/H]) with coefficients in Z. In homological degree 0, this recovers the isomorphism (13).

3.3

Pointed G-sets.

To obtain more information about derived Mackey functors, it is convenient to use the following observation. Say that a diagram (1) is restricted if the map l is injective, and note that the pullback of an injective map is an injective map. Therefore we have a subcategory QI ΓG ⊂ QΓG whose maps are isomorphism classes of restricted diagrams, and a subcategory SI ΓG ⊂ ΓG whose maps are pairs g, α with injective αn . Say that a map in SI ΓG is special if it is special in SΓG , say that an object E ∈ D(SI ΓG , R) is special if it inverts all special maps, and let DSI (ΓG , R) ⊂ D(SI ΓG , R) be the full subcategory spanned by special objects. Then the projection q : SΓG → QΓG restricts to a projection q : SI ΓG → QI ΓG , and we have a natural functor (25)

q ∗ : D(QI ΓG , R) → DSI (ΓG , R).

However, unlike the functor (20), this functor is an equivalence of categories (this is [5,Corollary 4.17]—roughly speaking, the reason this holds is that restricted diagrams (1) have no non-trivial automorphisms).

Derived Mackey functors and profunctors: an overview of results

The category QI ΓG is naturally identified with the category ΓG+ of finite pointed G-sets (that is, finite G-sets S with distinguished Ginvariant element o ∈ S). The equivalence sends S to S \ {o}, and a map f : S → S goes to the diagram f

S \ {o} ←−−−− S \ f −1 ({o }) −−−−→ S \ {o }. The equivalence (25) together with the restriction with respect to the embedding SI ΓG → SΓG then provide a natural functor (26)

r : DS(ΓG , R) → D(ΓG+ , R),

and we introduce the following definition. Definition 3.3.1 ([5, Definition 6.9]). For any derived Mackey functor E ∈ DM(G, R) and any simplicial finite pointed G-set X : ∆o → ΓG+ , the homology complex of X with coefficients in E is given by C (X, E) = C (∆o , X ∗ r(E)),

where r is the restriction functor (26), and C (∆o , −) is the homology complex of the small category ∆o . Moreover, in [5, Subsection 7.4], this definition is extended in the following way: for any E ∈ DM(G, R) and X ∈ ∆o ΓG+ , one defines a product X ∧ E ∈ DM(G, R). This product is functorial in X and E. For any S ∈ ΓG , one has a natural identification (27)

(X ∧ E)(S) ∼ = C (X ∧ S+ , E),

where S+ ∈ ΓG+ is obtained by adding a distinguished element o to the set S, and X ∧ S+ stands for pointwise smash-product of pointed sets. With this product, one can prove the following analog of (15). Definition 3.3.2. A simplicial finite pointed G-set X ∈ ∆o ΓG+ is adapted to a normal subgroup N ⊂ G if 1. X N = [1]+ is the constant pointed simplicial set with two elements, one distinguished, one not, and 2. for any subgroup H ⊂ G not containing N , the reduced chain homology complex C (X H , Z) is acyclic.

D. Kaledin

Lemma 3.3.3 ([5, Lemma 7.14]). Assume given a finite pointed simplicial G-set XN ∈ ∆o ΓG+ adapted to a normal subgroup N ⊂ G. Then for any E ∈ DM(G, R), we have a natural isomorphism X ∧E ∼ = InflN (ΦN (E)). We note that together with (27), Lemma 3.3.3 provides a canonical identification (28)

ΦN (E)(S) ∼ = C (XN ∧ S+ , E)

for any S ∈ ΓW , W = G/N . It is not difficult to show that for any N ⊂ G, an adapted set XN ∈ ∆o ΓG+ does exist. For example, one can take the union SN of all G-orbits [G/H], H ⊂ G not containing N , consider the pointed simplicial set E(SN ) given by n+1 , E(SN )([n]) = SN

(29)

and let XN be the cone of the natural map E(XN ) → [1]+ . If for this choice of XN , one computes the homology of the simplicial category ∆o by the standard complex, then (28) gives (15) in homological degree 0. Definition 3.3.4. A simplicial finite pointed G-set X ∈ ∆o ΓG+ is a homological sphere if for any subgroup H ⊂ G, we have C (X H , Z) ∼ = Z[dH ] for some integer dH ≥ 0. Lemma 3.3.5 ([5, Proposition 7.18]). For any homological sphere X, the functor E → X ∧ E is an autoequivalence of the category DM(G, R). Constructing homological spheres is as easy as constructing adapted sets. For example, for any finite-dimensional real representation V of the group G, we can take a G-invariant triangulation of its one-point compactification SV and obtain a homological sphere (the dimensions dH are given by dH = dimR V H ). Thus Lemma 3.3.5 produces many autoequivalences of the category DM(G, R).

3.4

An example.

To illustrate the theory of derived Mackey functors on a concrete example, let us consider the case G = Z/pZ, the cyclic group of a prime

Derived Mackey functors and profunctors: an overview of results

order. Then there are exactly two subgroups in G, the trivial subgroup {e} ⊂ G and G itself, so that up to an isomorphism, OG has two objects: the free orbit [G/{e}] and the trivial orbit [G/G]. Thus a G-Mackey functor E ∈ M(G, R) is defined by two R-modules, E 0 = E([G/{e}]) and E 1 = E([G/G]). The module E 0 carries the action of the group G. Equivalently, if we denote by σ ∈ Z/pZ the generator, then we have an automorphism σ : E 0 → E 0 such that σ p = id. Moreover, there one non-trivial map f : [G/{e}] → [G/G]. Since f ◦σ = f , the corresponding maps f∗ and f ∗ induce natural maps (30)

V = f∗ : (E 0 )σ → E1 ,

F = f ∗ : E 1 = (E 0 )σ ,

where (E 0 )σ , (E 0 )σ stands for coinvariants and invariants with respect to the automorphism σ. The double coset formula (6) then reads as (31)

F ◦ V = id +σ + · · · + σ p−1 .

These are the only conditions: the category M(G, R) is equivalent to the category of pairs E 0 , E 1 of two R-modules equipped with an automorphism σ : E 0 → E 0 of order p and two maps (30) satisfying (31). To obtain a similar description of the category DM(G, R), choose a resolution P of the trivial Z[G]-module Z by finitely generated projective Z[G]-modules, and denote C (G, −) = P ⊗Z[G] −,

C (G, −) = HomZ[G] (P , −)

the homology and cohomology complexes of the group G computed using P (for example, one can take the standard periodic resolution, and this would give the standard periodic complexes). For any R[G]-module E, we have H0 (G, E) = Eσ and H 0 (G, E) = E σ , and the functorial trace map id +σ + · · · + σ p−1 : Eσ → E σ induces a functorial trace map (32)

tr : C (G, −) → C (G, −).

Then DM(G, R) is obtained by inverting quasiisomorphisms in the category of pairs E 0 , E 1 of complexes of R-modules equipped with an automorphism σ : E 0 → E 0 of order p and two maps

(33)

C (G, E 0 ) −−−−→ E 1 −−−−→ C (G, E 0 ) V

whose composition F ◦ V coincides with the trace map (32), F ◦ V = tr. This description shows clearly why the category DM(G, R) is different

D. Kaledin

from the derived category D(M(G, R)). Indeed, objects in D(M(G, R)) are also represented by pairs E 0 , E 1 and maps σ, F , V , but the homology and cohomology complexes in (33) are replaces with coinvariants (E 0 )σ and invariants (E 0 )σ . To make these quasiisomorphic to the whole homology complexes, one needs to choose a representative E 0 that is both h-projective and h-injective as a complex of R[G]-modules. For general complex of R[G]-modules, such a representative does not exists. For any object E ∈ DM(G, R) represented by a pair E 0 , E 1 as above, the geometric fixed points ΦG (E) ∈ D(R) can be computed by 1 Lemma 3.3.3. The result is, ΦG (E) is quasiisomorphic to the cone E of the map V of (33). This suggests a very useful alternative description of the category DM(G, R). Namely, recall that the Tate cohomology complex Č (G, E ) with coefficients in a complex E of R[G]-modules is by definition the cone of the trace map (32). Then the map F in (33) induces a natural map

ϕ : E → Č (G, E 0 ). 1

(34)

Conversely, given E and the map ϕ, one recovers E 1 as the cone of the natural map 1

E −−−−→ Č (G, E 0 ) −−−−→ C (G, E 0 )[1], 1

this E 1 comes equipped with the map V of (33), and ϕ then induces the map F . We see that DM(G, R) can be obtained by inverting quasiiso1 morphisms in the category of pairs E 0 , E of complexes of R-modules equipped with an automorphism σ : E 0 → E 0 of order p and a map ϕ of (34). To rephrase this description even further, let P be the cone of the augmentation map P → Z, where P is our fixed projective resolution. Then one can always choose E 0 to be an h-injective complex of R[G] modules, and in this case, Č (G, E 0 ) is quasiisomorphic to the sum-total complex of the bicomplex (P ⊗ E 0 )σ .

Then the map ϕ of (34) becomes a σ-invariant map ϕ : E → P ⊗ E 0 , 1

Derived Mackey functors and profunctors: an overview of results

and the whole data E 0 , E , σ, ϕ can be packaged into a single DG comodule over the DG coalgebra T (G, R) over R of the following form: 1

(35)

T (G, R) =

R[G] P ⊗ R , 0 R

where R[G] is the group coalgebra of the group G, and P ⊗R is the R[G]comodule obtained from P . The category DM(G, R) is then equivalent to the derived category D(T (G, R)) (that is, the category obtained by inverting quasiisomorphisms in the category of DG comodules over T (G, R), as in [4, Subsection 1.5.3]). Using DG coalgebras here is essential – the complex P is infinite, and one cannot simply dualize things and interpret D(T (G, R)) as the derived category of DG modules over a DG algebra. Let us also note that the complex P ⊗R is acyclic, so that T (G, R) is quasiisomorphic to the diagonal DG coalgebra T (G, R) with entries R[G] and R. However, the derived categories D(T (G, R)) and D(T (G, R)) are different: the latter is the direct sum of D(R) and D(R[G]), while the former is obtained by gluing these two categories along the gluing functor given by Tate cohomology.

3.5

Maximal Tate cohomology.

To extend the description of derived Mackey functors in terms of geometric fixed points and Tate cohomology to arbitrary finite groups, note that for any group G and derived Mackey functor E ∈ DM(G, R), the value E([G/{e}]) of E at the free orbit [G/{e}] ∈ ΓG is acted upon by G = AutG ([G/{e}]) and gives an object in the derived category D(R[G]). For any subgroup H ⊂ G with WH = NH /H, we can apply this observation to the group WH . Then sending E ∈ DM(G, R) to ΦH (E)([WH /{e}]) ∈ D(R[WH ]) gives a natural functor Φ

: D(G, R) → D(R[WH ]),

a restricted version of the geometric fixed points functor of (23). It has been proved in [5, Lemma 7.2] that this functor has a right-adjoint restricted inflation functor (36)

Infl

: D(R[WH ]) → D(G, R).

D. Kaledin

Taking the direct sum of the functors Φ over all conjugacy classes of subgroups H ⊂ G, one obtains the functor Φ : DM(G, R) → D(R[WH ]), (37) H⊂G

and one proves the following result.

Lemma 3.5.1 ([5, Lemmas 7.1, 7.3]). The functor Φ of (37) is conservative, and for any H ⊂ G, the restricted inflation functor (36) is fully faithful. Moreover, let DMH (G, R) ⊂ DM(G, R) be the image of H the fully faithful embedding Infl . Then every object E ∈ DM(G, R) is an iterated extensions of objects EH ∈ DMH (G, R), H ⊂ G, and we have Hom(EH , EH ) = 0 unless H is conjugate to a subgroup in H . Thus the category DM(G, R) admits a semiorthogonal decomposition whose graded pieces DMH (G, R) are numbered by conjugacy classes of subgroup H ⊂ G, and we have DMH (G, R) ∼ = D(R[WH ]). For any two subgroups H ⊂ H ⊂ G, we then have the gluing functor (38)

H EH ◦ Infl = Φ

: D(R[H ]) → D(R[H]).

To compute these gluing functors, one introduces in [5, Subsection 7.2] a certain generalization of Tate cohomology that we call maximal Tate cohomology. Let us describe it. Recall that for any subgroup H ⊂ G, the restriction functor G rH : D(R[G]) → D(R[H])

has a left and right-adjoint induction functor iH G : D(R[H]) → D(R[G]). Definition 3.5.2 ([5, Definition 7.4]). A Z[G]-module E is induced if E is a direct summand of a sum of objects of the form iH G (EH ), H ⊂ G a proper subgroup, EH a finitely generated Z[H]-module projective over Z. For any E ∈ Db (Z[G]), the maximal Tate cohomology Ȟ (G, E) is given by Ȟ (G, E) = RHomD(Z[G]) (Z, E), where Dib (Z[G]) ⊂ Db (Z[G]) is the smallest Karoubi-closed triangulated subcategory containing all induced modules, and D(Z[G]) = Db (Z[G])/Dib (Z[G]) is the quotient category.

Derived Mackey functors and profunctors: an overview of results

If G = Z/pZ, a Z[G]-module is induced if and only if it is finitely generated and projective, so that maximal Tate cohomology coincides with the usual Tate cohomology. In general, they are different. For example, if G = Z/nZ is a cyclic group, then Ȟ (G, −) = 0 unless n is a prime (this is [4, Lemma 7.15 (ii)]). To compute maximal Tate cohomology, and to generalize it to possibly infinite coefficients, it is convenient to introduce the following. Definition 3.5.3. A complex P of Z[G]-modules is maximally adapted if 1. Pi = 0 for i < 0, P0 ∼ = Z, and Pi is induced for any i ≥ 1, and

H (P ) is contractible for any proper subgroup H ⊂ G. 2. rG

Then for any ring R and maximally adapted complex P , one defines the Tate cohomology object with coefficients in E ∈ D(R[G]) as (39)

Č (G, E) = lim C (G, E ⊗ F l P ), l

→

where F l P stands for the stupid filtration. It has been proved in [5, Subsection 7.2] that as an object in D(R), this does not depend on the choice of P , and for any E ∈ Db (Z[G]), the complex Č (G, E) computes the maximal Tate cohomology groups Ȟ (G, E) of Definition 3.5.2. If G is a normal subgroup in a larger group G , then for any E ∈ D(R[G ]) and an appropriate choice of an adapted complex, the expression (39) also defines Č (G, E) as an object in D(R[G /G]). A full description of the gluing functors (38) in terms of the maximal Tate cohomology objects (39) is given in [5, Proposition 7.10]. Here we only reproduce the answer in the key case H = {e}, H = G. In this case, we have G ∼ E{e} = Č (G, −).

Based on this, in [4, Section 6], one develops a description of the category DM(G, R) in terms of a certain upper-triangular DG coalgebra T (G, R) similar to the coalgebra (35). Unfortunately, this is rather heavy technically (in particular, one cannot get a genuine DG coalgebra and has to settle for its A∞ -version). A big simplification occurs in the case of the cyclic group G = Z/nZ, n ≥ 1. As we have mentioned, vH (Z/nZ, −) = 0 unless n is a prime; moreover, for any two primes p, p , it has been proved in [5, Lemma 9.10] that Č (Z/pZ, Č(Z/p Z, −)) = 0,

D. Kaledin

so that the compositions of the gluing functors (38) vanish tautologically. Because of this, it has been possible to obtain a reasonably simple description of DM(G, R). Let us reproduce it (this is [5, Subsection 9.4]). Fix an integer n ≥ 1, let G = Z/nZ, and denote by In the groupoid of G-orbits. These are numbered by divisors of n, so that explicitly, we have In = ptd , d|n

where ptd stands for the groupoid with one object with automorphism group Z/dZ. For any prime p, let Inp = ptd ⊂ In ,

p|d|n

and let In be the disjoint union of Inp over all primes p. Of course Inp is empty unless p divides n. For any p dividing n, we have the natural embedding i : Inp → In , and we also have a natural projection π : Inp → In given by the union of quotient maps ptpm → ptm , m a divisor of n/p. Let (40)

i, π : In → In

be the disjoint union of these functors. Finally, choose a projective resolution P of the constant functor Z ∈ Fun(In , Z), and let P be the cone of the augmentation map P → Z.

Definition 3.5.4. An R-valued fixed points datum for In is a pair M , α of a complex M in the category Fun(In , R) and a map α : π ∗ M → P ⊗ i∗ M ,

where i and π are the projections (40).

Then In -fixed points data form a category, and inverting quasiisomorphisms in this category, one obtains a category Dnα (R). It has been shown in [5, Subsection 9.3] that Dnα (R) is a triangulated category that does not depend on the coice of a resolution P . One then proves the following comparison result. Proposition 3.5.5 ([5, Proposition 9.14 (i)]). For any n ≥ 1 and ring R, there exiss a natural equivalence of triangulated categories DM(Z/nZ, R) ∼ = Dnα (R).

Derived Mackey functors and profunctors: an overview of results

Mackey profunctors.

4.1

Definitions.

Assume now given an infinite group G. Then if one considers arbitrary G-sets, all the theory of Section 2 and Section 3 becomes trivial (for example, the Burnside ring of (12) would be identically zero). If we stick to finite G-sets, the theory works, up to a point. However, finer parts of the theory such as Subsection 3.5 break down. As it turns out, there is an interesting possibility in between the two extremes. It is based on the following definition. Definition 4.1.1 ([5, Definition 3.1]). A G-set S is admissible if 1. for any s ∈ S, its stabilizer Gs ⊂ G is a cofinite subgroup, and 2. for any cofinite subgroup H ⊂ G, the fixed points set S H is finite. Explicitly, any G-set S decomposes into a disjoint union of orbits [G/Hs ]. (41) S= s∈S/G

Then S is admissible iff all the subgroups Hs ⊂ G are cofinite, and for any cofinite subgroup H ⊂ G, at most a finite number of them contain H. G , and one notes One denotes the category of admissible G-sets by Γ that this category has fibered products. Therefore we can consider the G and the embedding (2). category Q Γ o

G , R) is additive Definition 4.1.2. For any ring R, an object E ∈ D(Γ if for any admissible G-set S with decomposition (41), the natural map E([G/Hs ]) E(S) → s∈S/G

is an isomorphism. An R-valued G-Mackey profunctor is an object E ∈ G , R) whose restriction e∗ E is additive. Fun(Q Γ We denote by

M(G, R) ⊂ Fun(QΓG , R)

the full subcategory spanned by Mackey profunctors. Since infinite sums in the category of R-modules are exact, M(G, R) is an abelian category.

D. Kaledin

As in the finite group case, for any cofinite subgroup H ⊂ G, we G and Γ H . The have the pair of adjoint functors ρH , γ H between Γ functors preserve fibered products, the corresponding functors Q(ρH ), Q(γ H ) are also adjoint, and the functor Q(ρH )! ∼ = Q(γ H )∗ preserves the additivity condition of Definition 4.1.2, thus sends Mackey profunctors to Mackey profunctors. The corresponding functor ΨH = Q(ρH )! ∼ R) → M(H, R) = Q(γ H )∗ : M(G,

is the categorical fixed points functor. Moreover, if we denote W = G → ΓW also preserves NH /H, then the fixed points functor ϕH : Γ fibered products, and we have an adjoint pair of functors (42)

G , R) → Fun(QΓW , R), ΦH = Q(ϕH )! : Fun(Q Γ G , R). InflH = Q(ϕH )∗ : Fun(QΓW , R) → Fun(Q Γ

For the same reasons as in the usual case, these preserve additivity, thus induce functors between M(G, R) and M(W, R), the geometric fixed points functor and the inflation functor. Lemma 2.2 also holds for Mackey profunctors. In particular, for any normal cofinite subgroup N ⊂ G, the inflation functor InflN is fully faithful. Any Mackey profunctor E gives rise to a Mackey functor EN = ΦN E ∈ M(W, R), and we have a natural surjective map E → InflN (EN ). For any two cofinite normal subgroups N ⊂ N ⊂ G, we have a natural isomorphism (43)

ΦN

EN ∼ = EN .

It is convenient to axiomatize the situation as follows.

Definition 4.1.3. An R-valued G-normal system E is a collection of Mackey functors EN ∈ M(G/N, R), one for each cofinite normal subgroup N ⊂ G, and a collection os isomorphisms (43), one for each pair of cofinite normal subgroups N ⊂ N ⊂ G. Normal systems form an additive R-linear category which we denote by N (G, R). Sending E to ΦN (E) gives a functor Φ : M(G, R) → N (G, R). This functor has a right-adjoint (44)

Infl : N (G, R) → M(G, R)

Derived Mackey functors and profunctors: an overview of results

sending a normal system EN to (45)

E = lim InflN (EN ), N

←

where the limit is taken over all cofinite normal subgroups N ⊂ G, with respect to the natural maps adjoint to the isomorphisms (43). It turns out that the following is true. Lemma 4.1.4 ([5, Proposition 3.5]). The functor Infl of (44) is fully faithful. Conversely, for any Mackey profunctor E ∈ M(G, R), the adjunction map (46)

E → Infl(Φ(E)) = lim InflN (ΦN (E)) N

←

is surjective. One says that a Mackey profunctor E is separated if the surjective map (46) is bijective – or equivalently, if E lies in the image of the fully s (G, R) ⊂ M(G, R) the full faithful embedding Infl. One denotes by M subcategory spanned by separated Mackey profunctors. This category is additive but not necessarily abelian. However, it is equivalent to N (G, R), and this allows to transport results about Mackey functors to separated Mackey profunctors. In particular, it has been proved in s (G, R) ⊂ Fun(Q Γ G , R) [5, Lemma 3.9] that the natural embedding M admits a left-adjoint additivization functor (47)

s (G, R). G , R) → M Add : Fun(Q Γ

Using this functor, one can extend the definition of the geomeric fixed points functor ΦH to an arbitrary subrgroup H ⊂ G. Indeed, even if H ⊂ G is not cofinite, the inflation functor InflH of (42) sends Mackey profunctors to Mackey profunctors, and it also sends separated Mackey profunctors to separated ones. Then it has a natural left-adjoint given by ΦH = Add ◦Q(ϕH )! . Another application of the addivization functor is tensor products of separated Mackey profunctors; these are defined by the same formula (11) as in the Mackey functor case. The product is associative, commutative and unital. The unit object is the Burnside Mackey profunctor Explicitly, it is given by A. H , A([G/H]) =A

D. Kaledin

H is given by (12) with ΓH replaced where the completed Burnside ring A H . By virtue of the decomposition (41), we have by Γ G = A

(48)

H⊂G

where the product is over conjugacy classes of cofinite subgroups H ⊂ G.

4.2

An example.

To illustate the difference between Mackey functors and Mackey profunctors, consider the case G = Z, the infinite cyclic group. Cofinite subgroups in Z are of the form nZ ⊂ Z, n ≥ 1, so that Z-orbits are numbered by positive integers. The Burnside ring AZ is given by (13); explicitly, as an abelian group, we have AZ = Z[ε1 , ε2 , . . . ], where the generators εn , n ≥ 1 correspond to Z-orbits [Z/nZ]. For any n, m ≥ 1, the product [Z/mZ] × [Z/nZ] is the union of copies of the orbit Z/{n, m}Z, where {n, m} is the least common multiple of n and m. Since the cardinality of this product is nm, this implies that (49)

εn εm =

nm ε , {n, m} {n,m}

and this completely defines the product in the Burnside ring AZ . Z , the sum (13) is replaced Now, for the completed Burnside ring A by the product (48). Therefore we have Z = Z{ε1 , ε2 , . . . }, A

that is, the group of infinite linear combinations of the generators εn , n ≥ 1. The product is still given by (49). Here is one observation we can make right away: the completed Z is isomorphic to the universal Witt vectors ring W(Z). Burnside ring A There is a reason for this coincidence, but it goes beyond the subject of the present paper. Another observation is the following. For any ring R, any R-valued Z-Mackey functor E is acted upon by the Burnside ring AZ , and any Z Z . Mackey profunctor E is acted upon by the completed Burnside ring A Assume that R is p-local for some prime p — that is, every integer n

Derived Mackey functors and profunctors: an overview of results

prime to p is invertible in R. Then for any such n, we have a well-defined endomorphism 1 Îľn : E â&#x2020;&#x2019; E, n

(50)

and (49) shows that this endomorphism is idempotent. Thus E is naturally equipped with a large family of commuting idempotents. If E is a Z-Mackey functor, then this is the end of the story. However, if E is a Z-Mackey profunctor, we can replace the commuting idempotents (50) with orthogonal commuting idempotents Îľ(n) given by 1 1 1 â&#x2C6;&#x2019; Îľi , Îľ(n) = Îľn Âˇ n i i does not divide n

where the product is over i prime to p. We have 1= Îľ(n) , n prime to p

R), we have a canonical decomposition so that for any E â&#x2C6;&#x2C6; M(Z, E(n) , E(n) = Im Îľ(n) â&#x160;&#x201A; E, (51) E= n prime to p

which is functorial in E, and gives a decomposition of the category R). M(Z, It turns out that the pieces of this decomposition can be described in terms of Mackey profunctors for the group Zp of p-adic integers. Namely, by definition, the category M(G, R) only depends on the profi is given nite completion of the group G. The profinite completion Z by = Z Zl , l

where the product is over all primes l. If we denote by Z p the product of = Zp Ă&#x2014; Z , and we have the geomeric all primes different from p, then Z p fixed points functor R) â&#x2020;&#x2019; M(Z R) â&#x2C6;ź Z, p , R). ÎŚ(p) = ÎŚZp : M(Z, = M(

In [5, Subsection 9.2], this functor has been refined â&#x20AC;&#x201C; for any integer n â&#x2030;Ľ 1 prime to p, one constructs a functor (p)

R) â&#x2020;&#x2019; M(Z p , R[Z/nZ]) ÎŚ[n] : M(Z,

D. Kaledin

(p)

such that Φ(p) = Φ(1) , and one proves the following result. Proposition 4.2.1 ([5, Proposition 9.4]). Assume that the ring R is p-local. Then the functor (p) R) → p , R[Z/nZ]) Φ(n) : M(Z, M(Z n prime to p

R), the compois an equivalence of categories, and for any E ∈ M(Z, (p) nent E(n) of the decomposition (51) corresponds to Φ(n) (E).

Z with the Witt vectors ring If one identifies the Burnside ring A W(Z), then the decomposition (51) corresponds to the p-typical decomposition of Witt vectors; because of this, in [5, Subsection 9.2], (51) is called p-typical decomposition.

4.3

Derived version.

The derived counterpart of the theory of Mackey profunctors is largely parallel to the theory of derived Mackey functors of Section 3. Since the G has fibered products, it can be plugged into the machinery category Γ G and the category of Subsection 3.1. We thus have the category S Γ G , R) for any ring R. The additivity condition of Definition 4.1.2 DS(Γ oG , R). A derived makes sense of objects in the derived category D(Γ G , R) whose base R-valued G-Mackey profunctor is an object E ∈ DS(Γ part E of (19) is additive. Denote the category of derived G-Mackey profunctors by DM(G, R). This category has a t-structure induced by the standard t-structure on G , R), and the heart of this t-structure is identified with M(G, DS(Γ R). For any cofinite subgroup H ⊂ G, we have the categorical fixed points functor ΨH : DM(G, R) → DM(H, R) defined exactly as in (22). Moreover, if we denote W = NH /H, then we have the adjoint pair R) → DM(W, R), ΦH : DM(G,

InflH : DM(W, R) → DM(G, R)

of the geometric fixed points functor ΦH and the inflation functor InflH defined as in (23). The inflation functor is defined even for a subgroup H ⊂ G that is not cofinite. For a cofinite normal subgroup N ⊂ G,

Derived Mackey functors and profunctors: an overview of results

the inflation functor InflN is fully faithful (this is [5, Lemma 6.12]). We also have a version of Lemma 3.3.3 (the statement is the same, but one needs to use a slightly stronger notion of an adapted set given in [5, Definition 6.3]). To proceed further, one needs to introduce a derived counterpart of the notion of a normal system. Simply repeating Definition 4.3.1 does not work since it does not produce a triangulated category – we have to package the same data in a more elaborate way. To do this, let N be the partially ordered set of normal cofinite subgroups N ⊂ G, ordered by inclusion, and let ΓG ⊂ ΓG × No

be the subcategory of pairs S, N such that N acts trivially on S (we treat the partially ordered set N as a small category in the usual way, and let No be the opposite category). For every N ∈ N, we have a natural functor τN : ΓG/N → ΓG sending S to S, N . Moreover, for any pair of cofinite normal subgroups N ⊂ N ⊂ G, we have the fixed points functor ϕN /N : ΓG/N → ΓG/N , and the inclusions S N /N ⊂ S glue together to give a map of functors τN ◦ ϕN

(52)

→ τN .

We also have the forgetful functor ν : ΓG → No sending S, N to N ∈ No , and to define derived normal systems, we do the S-construction of Subsection 3.1 relatively over No – that is, we consider the full subcategory S(ΓG / No ) ⊂ SΓG

formed by diagrams (16) in ΓG such that all maps become invertible after applying ν. Then for any N ∈ No , the functor τN gives a functor S(τN ) : SΓG/N → S(ΓG / No ), and for a pair N ⊂ N ⊂ G, N, N ∈ N, the morphism (52) gives a morphism (53)

S(τN ) → S(τN ) ◦ S(ϕN

Therefore any object E ∈ D(S(ΓG / N), R) produces a collection of objects EN = S(τN )∗ E ∈ D(SΓG/N , R), N ∈ N,

related by morphisms (54)

EN → S(ϕN

) ∗ EN .

D. Kaledin

Definition 4.3.1. An R-valued derived G-normal system is an object E ∈ D(S(ΓG / N), R) such that for any N ∈ N, EN ∈ D(SΓG/N , R) is a Mackey functor, and for any N ⊂ N ⊂ G, the map ΦN /N EN → EN adjoint to the map (54) is an isomorphism. By construction, derived normal systems form a triangulated category; we denote it by DN (G, R). For every integer n, we let DN ≤n (G, R) ⊂ DN (G, R) be the full subcategory formed by objects E such that for any N ∈ N, EN lies in DM≤n (G/N, R). These subcategories do not necessarily give a t-structure on DN (G, R). However, they are perfectly well-defined. With these definitions, one constructs a functor Φ : DM(G, R) → DN (G, R) such that for any E ∈ DM(G, R) and N ∈ N, Φ(E)N is canonically identified with ΦN (E), and the map (43) is adjoint to the map (54). Then one proves the following derived counterpart of Lemma 4.1.4. Proposition 4.3.2 ([5, Proposition 8.2]). Assume that the group G is finitely generated. Then for every integer n, the functor Φ induces an equivalence of categories Φ : DM≤n (G, R) ∼ = DN ≤n (G, R). As a corollary of this Proposition, we can consider the union DN − (G, R) = DN ≤n (G, R) ⊂ DN (G, R) n

of all the categories DN ≤n (G, R), and see that it is naturally equiv− (G, R) ⊂ DM(G, R) of derived alent to the full subcategory DM Mackey profunctors bounded from above with respect to the standard t-structure. As we see, Proposition 4.3.2 is much stronger that Lemma 4.1.4: a derived Mackey profunctor is separated as soon as it is bounded from above. In particular, any Mackey profunctor E ∈ M(G, R) ⊂ DM(G, R) is separated in the derived sense. What happens is, the equivalence Infl inverse to Φ is given explicitly by

Infl(E) = lim InflN (EN ), N

←

Derived Mackey functors and profunctors: an overview of results

but since the inverse limit is not an exact functor, this is different from (45) – there could be non-trivial contributions from R1 lim← in the righthand side. One shows that we in fact have Ri lim← = 0 for i ≥ 2, so for any Mackey profunctor E, the kernel of the canonical surjective map (46) is identified with R1 lim InflN (L1 ΦN (E)). N

←

Applications of Proposition 4.3.2 are similar to those of Lemma 4.1.4. Firstly, one proves in [5, Lemma 8.3] that the embedding −

G , R) (G, R) ⊂ DS − (Γ DM

admits a left-adjoint additivization functor (55)

−

G , R) → DM (G, R). Add : DS − (Γ

We note that by adjunction, Add is right-exact with respect to the standard t-structures, and on the hearts of the standard t-structures, it G , R) → M(G, induces a functor Fun(Q Γ R) right-adjoint to the natural embedding — that is, a refinement of the additivization functor (47). Using this refinement, one extends the product (11) to all Mackey profunctors, not only the separated ones, and one does the same with the geometric fixed points functors ΦH for arbitrary subgroups H ⊂ G. In the derived theory, one uses (55) to define the tensor product of derived Mackey profunctors by (24), and one defines the geometric fixed points functor ΦH with respect to an arbitrary subgroup H ⊂ G by ΦH = Add ◦S(ϕH )! .

This is left-adjoint to the inflation functor InflH . We also have a version of Lemma 3.3.5 — namely, [5, Lemma 8.7] — and can prove an analog of Lemma 3.5.1, although the semiorthognal decomposition on DM− (G, R) would have an infinite number of terms. Finally, let us note that in the case G = Z, the infinite cyclic group, we have a complete analog of Proposition 3.5.5. Namely, let I= ptn n≥1

be the groupoid of Z-orbits, let I p ⊂ I be the subcategory spanned by ptnp , n ≥ 1, let I be the disjoint union of the categories I p over all prime p, and let (56)

i, π : I → I

D. Kaledin

be the natural functors defined in the same way as (40). Choose a projective resolution P of the constant functor Z ∈ Fun(I , Z), and let P be the cone of the augmentation map P → Z. Definition 4.3.3. An R-valued fixed points datum is a pair M , α of a complex M in the category Fun(I, R) bounded from above, and a map α : π ∗ M → P ⊗ i∗ M ,

where i and π are the projections (56).

Then just as in the case of a finite cyclic group, I-fixed points data form a category, and inverting quasiisomorphisms in this category, one obtains a category Dα (R). One shows that Dα (R) is a triangulated category that does not depend on the choice of a resolution P , and proves the following result. Proposition 4.3.4 ([5, Proposition 9.14 (ii)]). For any ring R, there exiss a natural equivalence of triangulated categories DM− (Z, R) ∼ = Dα (R). Acknowledgement I am grateful to the organizers of the International School on TQFT, Langlands and Mirror Symmetry in Playa del Carmen, Mexico, in March 2014 for inviting me and giving me the opportunity to present my results. It is a pleasure to submit this overview to the proceedings. D. Kaledin Steklov Math Institute, Algebraic Geometry section, Laboratory of Algebraic Geometry, NRU HSE, and, IBS Center for Geometry and Physics, Pohang, Rep. of Korea, kaledin@mi.ras.ru

References [1] T. tom Dieck, Transformation groups, De Gruyter, Berlin-New York, 1987.

Derived Mackey functors and profunctors: an overview of results

[2] A.W.M. Dress, Contributions to the theory of induced representations, in Algebraic K-Theory II, (H. Bass, ed.), Lecture Notes in Math. 342, Springer-Verlag, 1973; pp. 183–240. [3] A. Grothendieck, SGA I, Exposé VI. [4] D. Kaledin, Derived Mackey functors, Mosc. Math. J. 11 (2011), 723–803. [5] D. Kaledin, Mackey profunctors, arxiv:1412.3248. [6] L.G. Lewis, J.P. May, and M. Steinberger, Equivariant stable homotopy theory, with contributions by J. E. McClure, Lecture Notes in Mathematics, 1213, Springer-Verlag, Berlin, 1986. [7] H. Lindner, A remark on Mackey functors, Manuscripta Math. 18 (1976), 273–278. [8] J.P. May, Equivariant homotopy and cohomology theory, with contributions by M. Cole, G. Comezana, S. Costenoble, A.D. Elmendorf, J.P.C. Greenlees, L.G. Lewis, Jr., R.J. Piacenza, G. Triantafillou, and S. Waner, CBMS Regional Conference Series in Mathematics, 91. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1996. [9] J. Thevenaz and P. Webb, The structure of Mackey functors, Trans. AMS 347 (1995), 1865–1961.

Morfismos, Vol. 22, No. 2, 2018, pp. 85–119 Morfismos, Vol. 22, No. 2, 2018, pp. 85–119

Perverse Schobers PerverseVadim Schobers Mikhail Kapranov Schechtman Mikhail Kapranov

Abstract

Vadim Schechtman

In this paper we introduce the notion of perverse schober and talk about how it is related to the understanding Abstract of Fukaya categories. In this paper we introduce the notion of perverse schober and talk 2010 Mathematics Subject Classification: 14A22, 14J33, 53D37. about how it is related to the understanding of Fukaya categories. Keywords and phrases: Perverse schobers, Fukaya categories. 2010 Mathematics Subject Classification: 14A22, 14J33, 53D37. Keywords and phrases: Perverse schobers, Fukaya categories.

Introduction

The 1 notion of a perverse sheaf, introduced in [7], has come to play a Introduction central role in algebraic geometry and representation theory. In particular, perverse sheavesinprovide “categorificaTheappropriate notion of acategories perverse of sheaf, introduced [7], has come to play a tions”central of various spaces, these spaces being recovered as parrole representation in algebraic geometry and representation theory. In the Grothedieck groups ofcategories the categories. ticular, appropriate of perverse sheaves provide “categorificaThe goalofofvarious this paper is to suggestspaces, the possibility of categorifying the as tions” representation these spaces being recovered very the concept of a perverse sheaf. In other words, we propose to develop Grothedieck groups of the categories. a theoryThe of perverse sheaves not vectorthespaces but of triangulated the goal of this paper is to of suggest possibility of categorifying categories. very concept of a perverse sheaf. In other words, we propose to develop Given a complex manifold X, annot analytic Whitney stratification S= a theory of perverse sheaves of vector spaces but of triangulated (Xα )categories. α∈A of X and a ground field k, one has the category Perv(X, S) of perverse sheaves of k-vector spaces onanalytic X smooth with respect to S. S = Given a complex manifold X, an Whitney stratification Traditionally, two field ways k, of one looking S):Perv(X, S) (Xα )α∈A there of X have and abeen ground has at thePerv(X, category of perverse sheaves of k-vector spaces on X smooth with respect to S. (1) Traditionally, General definition: an abelian in Perv(X, the trianguthere haveasbeen two wayssubcategory of looking at S): b lated category Dconstr (X, S) of constructible complexes of sheaves of spaces on X, smooth respect to S. (1)k-vector General definition: as an with abelian subcategory in the triangub lated category D (X, S) of constructible complexes of sheaves (2) Quiver description constr (for some particular (X, S)): as a category of k-vector spaces on X, smooth with respect to S. of diagrams of some given type formed by vector spaces (Vi )i∈I and maps betweendescription them subject(for to some certain relations.(X, These (2) Quiver particular S)): diagrams as a category have of the following diagrams of features: some given type formed by vector spaces (Vi )i∈I and maps between them subject to certain relations. These diagrams have the following features: 85 85

Mikhail Kapranov and Vadim Schechtman

V having the same ends but (2a) Arrows come in pairs Vi j opposite directions. This reflects the (Verdier) self-duality of Perv(X, S).

(2b) In most cases, the relations contain 2 or 3 summands, with coefficients ±1.

So far, there is no obvious direct way to categorify the approach (1) since it is not clear what are complexes of triangulated categories. On the other hand, we observe that the features (2a) and (2b) are of the kind that immediately suggest a categorical generalization. We can replace vector spaces Vi by triangulated categories Vi and arrows by exact functors. The pairs of opposite arrows in (2a) can be interpreted as adjoint pairs of functors, 2-term relations as isomorphisms of functors and 3-term ones as exact triangles in appropriate functor categories (see Appendix for the precise framework in which these make sense). By forming Grothendieck groups Vi = K0 (Vi ) ⊗ k1 of such a diagram of categories, one would then obtain a quiver in the original sense, i.e., a perverse sheaf. The idea that the cone can be seen as a categorical analog of the difference, lies, of course, at the very foundations of algebraic Ktheory, especially in the Waldhausen approach. This strongly suggests that there should be meaningful objects which can be understood as “perverse sheaves of triangulated categories" and which give usual perverse sheaves by passing to the Grothendieck groups. We propose to call such hypothetical objects perverse Schobers (or, sometimes, for brevity, simply Schobers), using the German analog2 of the English word “stack" which would be the correct (but overused) term for speaking of “sheaves of categories". In this paper we work out several basic examples of quiver descriptions of perverse sheaves and define, in an ad hoc way, what should be the perverse Schobers in these situations. In the simplest case, we propose, in §2, to identify perverse Schobers on a disk with one allowed singular point, with spherical functors of [1][4]. For a disk with several allowed singular points we propose, in §3, a definition in terms of certain diagrams of spherical functors, and explain the invariance properties of such a definition. Among other things, we reformulate the classical Picard-Lefschetz formula as a general statement 1

or apply any other functor from the category of triangulated categories to kvector spaces, for example, the higher Quillen K-theory, or Hochschild cohomology 2 A literal Russian analog would be the word стог. We learned the term “Schober" from W. Soergel.

Perverse Schobers

about perverse sheaves on a disk, and then lift it to a distinguished triangle associated to a perverse Schober and a certain configuration of paths. Such “Picard-Lefschetz triangles" should therefore be considered as fundamental features of perverse Schobers. It is natural to expect analogous features in the case dim(X) > 1 and even consider them as “codimension 1 data" of a perverse Schober. A series of examples of spherical functors is provided by representation theory. More precisely, for a reductive group G we have spherical functors acting in the derived category of sheaves on G/B and satisfying the relations of the corresponding braid group Br(g). In §5 we review these examples and suggest a conjectural interpretation in terms of perverse Schobers on h/W . Most of the known quiver descriptions of Perv(X, S) can be obtained using a choice of “cuts" which are certain totally real subvarieties K ⊂ X (of real dimension equal to dimC X). In §6 we summarize the features of such cuts and note that Lagrangian varieties, used in constructing Fukaya categories [18][34], provide a reasonable class of candidates for cuts. One can therefore expect Fukaya-categorical constructions to have a bearing on the problem of classification of perverse sheaves. Additionally, we discuss the idea of defining “Fukaya categories with coefficients". This idea was proposed by M. Kontsevich in order to study the usual Fukaya category of a manifold by fibering it over a manifold of smaller dimension. We suggest that perverse Schobers should be considered as the right “coefficient data" for such a definition, just like sheaves are natural coefficient data for defining cohomology. We would like to thank A. Bondal, V. Ezhov, M. Finkelberg, D. Nadler, P. Schapira, W. Soergel, Y. Soibelman and B. Toën for useful discussions and correspondence. V.S. is grateful to the Kavli IPMU for hospitality and support during the visits when this paper was finished. The work of M. K. was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.

Perverse Schobers on a disk: spherical functors

A. Disk with one marked point. Let ∆ be the unit disk in C and Perv(∆, 0) be the category of perverse sheaves on ∆ with the only possible singularity at 0. The most iconic example of a quiver description of perverse sheaves is the following classical statement [6] [19].

Mikhail Kapranov and Vadim Schechtman

Theorem 2.1. Perv(â&#x2C6;&#x2020;, 0) is equivalent to the category P1 of quadruples (ÎŚ, Î¨, u, v) where ÎŚ, Î¨ â&#x2C6;&#x2C6; Vectk and (1)

ÎŚ

v u

Î¨

are linear maps such that (2)

TÎ¨ := IdÎ¨ â&#x2C6;&#x2019; vu is an isomorphism.

Exercise 2.2. Show that TÎ¨ is an isomorphism iff TÎŚ : IdÎŚ â&#x2C6;&#x2019; uv is an isomorphism. â&#x2C6;&#x2020;

Figure 1: Defining ÎŚ and Î¨ topologically. One way of constructing an explicit equivalence is as follows [19]. Choose a base point b on the boundary of â&#x2C6;&#x2020; and connect it with 0 by a simple arc K, see Fig. 1. Then to F â&#x2C6;&#x2C6; Perv(â&#x2C6;&#x2020;, 0) we associate the spaces ÎŚ(F) := H1K (F)0 H1K (â&#x2C6;&#x2020;, F) (vanishing cycles), Î¨(F) := Fb H1K (F)b

(nearby cycles).

(We recall that F|â&#x2C6;&#x2020;â&#x2C6;&#x2019;{0} is a local system in degree 0). The map v = vF is the generalization map [20][15] for the constructible sheaf H1K (F) on K, and u is the composition (3)

counterclockwise continuation

H 0 (â&#x2C6;&#x2020; â&#x2C6;&#x2019; K, F)

Î´

H1 (â&#x2C6;&#x2020;, F). K

Remark 2.3. Further, we have the following elementary statements which we recall here as indicative of a certain pattern. (1) HiK (F) = 0 for i = 1.

Perverse Schobers

(2) The sheaf R(F) = H1K (F) on K is constant on K − {0} so it has only two essentially different stalks R(F)0 = Φ(F) and R(F)b = Ψ(F). (3) Each of the two stalks, considered as a functor Perv(∆, 0) → Vectk , is an exact functor which takes Verdier duality to vector space duality. (4) The map u = uF is the dual uF = (vF ∗ )∗ . B. Spherical functors. As a natural categorical analog of the data (1)(2) we would like to suggest the following remarkable concept introduced by R. Anno [1]. Definition 2.4. Let S : D0 −→ D1

be an exact functor between triangulated categories (see Appendix for conventions). Assume that S admits a left adjoint L and a right adjoint R, so that we have the unit and counit natural transformations SR ⇒ IdD1 , IdD0 ⇒ RS,

LS ⇒ IdD0 ,

IdD1 ⇒ SL,

whose cones will be denoted by T1 = Cone{SR ⇒ IdD1 },

T1 = Cone{IdD1 ⇒ SL}[−1] (the twist functors), T0 = Cone{IdD0 ⇒ RS}[−1],

T0 = Cone{LS ⇒ IdD0 } (the cotwist functors).

We call S a spherical functor, if: (SF1) T1 is an equivalence. (SF2) The composition R → RSL → T0 L[1] is an isomorphism. In other words, composition with T0 identifies R and L. In this case T1 is quasi-inverse to T1 and T0 is quasi-inverse to T0 . More precisely, each of the pairs of adjoint functors (4)

S R

D , 1

S L

Mikhail Kapranov and Vadim Schechtman

can be regarded as an analog of (1). Further, (SF1) is an analog of (2), the adjunction unit allowing us to take the “categorical difference", i.e., the cone. Further still, the categorical analog of Exercise 2.2 can be found in the following result of Anno and Logvinenko [4]. Theorem 2.5. In addition to (SF1) and (SF2) consider the following two conditions: (SF3) T0 is an equivalence. (SF4) The composition LT1 [−1] → LSR → R is an isomoprhism. Then, any two of the conditions (SF1)-(SF4) imply the other two. So we can consider a diagram (4), i.e., the data of a spherical functor, as the data defining a “perverse Schober” over (∆, 0). By passing to K0 (or to any Ki , or to any homological functor, Hochschild homology for example) we get a perverse sheaf over (∆, 0). B. Examples of spherical functors. We now give some examples, to be used later. Example 2.6. Let Sd be the d-dimensional sphere and q : Sd → pt be the projection. We then have the functor S=q −1

D0 := Db (Vect) = Db (pt) −→ Db (Sd ) =: D1 with right adjoint R = Rq∗ and left adjoint L Rq∗ [−d]. The second adjunction is the Poincaré duality. Formaly, it comes from the adjoint pair (Rq! , q ! ) by noticing that q ! q −1 [d] (since q is smooth orientable of relative dimension d) and that Rq! = Rq ∗ (since q is proper), see [25] for background. More intrinsically, L is canonically identified with the tensor product of Rq∗ [−d] and H d (Sd , k), the 1-dimensional k-vector space spanned by global orientations of Sd . Proposition 2.7. (a) S is a spherical functor. (b) More generally, for any Sd -fibration q : Z → Y of CW-complexes, the functor S=q −1

D0 := Db (Y ) −→ Db (Z) =: D1 is a spherical functor.

Perverse Schobers

Proof: We prove (a), since (b), being a relative version, is proved in the same way. The functor T1 = Cone{SR ⇒ IdD1 } is the endomorphism of Db (S d ) defined as follows. Let j

U = (Sd × Sd ) − ∆ → Sd × Sd be the embedding of the complement of the diagonal, π1 , π2 : Sd × Sd → Sd be the projections and p1 , p2 : U → Sd be their restrictions to U . Then T1 (F) = Rp2! (p∗1 F), a formula remindful of the Fourier-Sato transform relating sheaves on dual spheres [32]. To see this, we write the functor in the RHS in terms of a “kernel", as Rπ2∗ ((π1∗ F) ⊗k K),

K = j! kU [1] ∈ Db (Sd × Sd )

and note the exact sequence 0 → K[−1] −→ kSd ×Sd −→ k∆ → 0, in which kSd ×Sd is the kernel for SR while k∆ is the kernel for IdD1 . The twist T1 = Cone{IdD1 ⇒ SL}[−1] can be found explicitly as T1 (F) = Rp2∗ (p∗2 F))[−1] = Rπ2∗ ((π1∗ F) ⊗k K ),

K = Rj∗ kU [−1].

Now, the condition (SF1), i.e., the fact that T1 and T1 are quasi-inverse to each other, can be established directly by finding the “composition” of the kernels K and K −1 −1 K ∗ K = Rπ13∗ (π12 K ⊗k π23 K ),

πij : Sd × Sd × Sd → Sd × Sd ,

and showing that both K ∗ K and K ∗ K are isomorphic to k∆ in degree 0. This amounts to the fact that for distinct points x, y ∈ Sd we have = 0, H • Sd − {x}, j! kSd −{x,y}

while for x = y we have H • (Sd − {x}) = k and Hc• (Sd − {x}) = k[−d]. Further, the cotwist T0 is the shift by (−d) tensored with H d (Sd , k), the 1-dimensional vector space of orientations of Sd . So it is an equivalence and (SF2) is also satisfied.

Mikhail Kapranov and Vadim Schechtman

Example 2.8. Note the particular case d = 2, when S2 = CP1 . Proposition 2.7 implies that for any P1 -fibration q : Z → Y of complex algebraic varieties, q −1 : Db (Y ) → Db (Z) is a spherical functor. Another class of examples is provided by quaternionic geometry, since HP1 = S4 . Let us now mention some “coherent” examples. Example 2.9. Recall that an n-dimensional smooth projective variety Z over k is called Calabi-Yau (in the strict sense), if H i (Z, OZ ) =

k, 0,

if i = 0, or i = n, otherwise.

Let X be a smooth algebraic variety over k, and q : Z → X be a smooth proper family of Calabi-Yau manifolds. Then the pullback functor q ∗ : b (X) → D b (Z) is a spherical functor. The proof is similar to that Dcoh coh of Proposition 2.7. Example 2.10. ([1]) Consider a diagram ρ

Y ←− D → X of smooth complex varieties X, D, Y , where ρ is a P1 -bundle and i an embedding of a divisor. We then have a diagram of adjoint functors L=ρ! i∗

D0 :=

b (Y Dcoh

)

←−

S=i∗ ρ∗

−→

R=ρ∗

b (X) Dcoh

←−

and R = T0 L where T0 = Cone{IdD0 ⇒ RS}. Lemma 2.11. (R. Anno, [1]). S is spherical iff the intersection index of D with a generic fiber of ρ is (−2). Example 2.12. A particular case of Example 2.10 and Lemma 2.11 is obtained for i ρ Y = pt ←− D = P1 → X = T ∗ P1 The corresponding spherical functor can be seen as a “quasi-classical approximation” to that in Example 2.8.

Perverse Schobers

Remark 2.13. In more precise terms, for a complex algebraic manib (T ∗ Z) can be thought of as fold Z, the coherent derived category Dcoh a quasi-classical approximation to the derived category Db (Z) of arbitrary sheaves on Z. Indeed, passing to solutions of D-modules gives a functor between derived categories in the first line of the following table (a functor that restricts to the Riemann-Hilbert equivalence between constructible and holonomic regular derived categories). It can be compared with the second line which is an instance of Serre’s theorem for the affine morphism p : T ∗ Z → Z. Coherent DZ -modules on Z Coherent p∗ OT ∗ Z = gr(DZ )modules on Z

(Arbitrary) sheaves on Z Coherent sheaves on T ∗ Z

Note that the functor S = q −1 on sheaf-theoretic derived categories in Example 2.8 matches, after being interpreted in terms of D-modules and passing to the associated graded, the functor S = i∗ ρ∗ on coherent derived categories in Example 2.12.

Disk with several marked points

A. Quiver description of perverse sheaves. Let B = {b1 , · · · , bn } be a finite set of marked points in the unit disk ∆ and Perv(∆, B) be the category of perverse sheaves on ∆ with possible singularities at B. We then have the following [21, Prop. 1.2]. Proposition 3.1. Perv(∆, B) is equivalent to the category Pn of diagrams formed by vector spaces Ψ, Φ1 , · · · , Φn and linear maps Φi

vi ui

such that each TΨ,i := IdΨ − vi ui is an isomorphism. The category Pn can be seen as an amalgamation of n copies of the category P1 from Theorem 2.1. To construct an equivalence, we choose a base point b ∈ ∂∆ and a “system of cuts" K, i.e., a set of simple arcs {K1 , · · · , Kn } with Ki connecting b with bi and with different Ki meeting only at a small common interval near b, see Fig. 2. Notationally, we view K as the union K = Ki ⊂ ∆.

Mikhail Kapranov and Vadim Schechtman

â&#x2C6;&#x2020; b1

.. .

Figure 2: Equivalence depending on a system of cuts.

to (5)

Given K, we have an equivalence FK : Perv(â&#x2C6;&#x2020;, B) â&#x2020;&#x2019; Pn sending F 1 ÎŚK i (F) = HK (F)bi ,

Î¨K (F) = Fb H1Ki (F)b = H1K (F)b .

Remark 3.2. We have the following elementary statements, continuing the pattern of Remark 2.3. (1) HiK (F) = 0 for i = 1. (2) Each stalk of R(F), considered as a functor Perv(â&#x2C6;&#x2020;, B) â&#x2020;&#x2019; Vectk , is an exact functor which takes Verdier duality to vector space duality. Remark 3.3. The space of cohomology with support H1K (â&#x2C6;&#x2020;, F) can be seen as â&#x20AC;&#x153;unitingâ&#x20AC;? all the spaces of vanishing cyclies ÎŚK i (F). As common in singularity theory, we can imagine that F obtained as a deformation of a perverse sheaf G â&#x2C6;&#x2C6; Perv(â&#x2C6;&#x2020;, 0) with only one (but more complicated) singular point at 0. Then H1K (â&#x2C6;&#x2020;, F) recovers ÎŚ(F). Note that a categorification of the space of vanishing cycles of an isolated singular point of a function is provided by the Fukaya-Seidel category [34], and the method of construction adopted in loc. cit. uses precisely a deformation into several Morse critical points. Therefore, the space H1K (â&#x2C6;&#x2020;, F) for a perverse sheaf F â&#x2C6;&#x2C6; Perv(â&#x2C6;&#x2020;, B), can be seen as a de-categorified analog of the Fukaya-Seidel category. The equivalence FK depends only on the isotopy class of K. Unlike the one point case, there are now many such classes, forming a set which we

Perverse Schobers

denote C. It is acted upon simply transitively by the Artin braid group Brn = s1 , Âˇ Âˇ Âˇ , snâ&#x2C6;&#x2019;1 si si+1 si = si+1 si si+1 . Indeed,

Brn = Ď&#x20AC;0 Diff + (â&#x2C6;&#x2020;; B, b)

is the group of isotopy classes of diffeomorphisms of â&#x2C6;&#x2020;, preserving orientation, preserving b as a point and B as a set. The equivalences FK for different K â&#x2C6;&#x2C6; C are connected by self-equivalences fĎ&#x192; of Pn : (6)

fĎ&#x192;

Pn ,

Ď&#x192; â&#x2C6;&#x2C6; Brn .

FĎ&#x192;(K)

Perv(â&#x2C6;&#x2020;, B) The self-equivalence fsi corresponding to a generator si of Brn , is given by [21, Prop. 1.3]: fsi Î¨, ÎŚj , ui , vj = Î¨, ÎŚ j , u j , vj , (7)

Î¨ = Î¨, ÎŚ j = ÎŚj , u j = uj , vj = vj ,

ÎŚ i+1 = ÎŚi , ÎŚ i = ÎŚi+1 ,

j = i, i + 1,

â&#x2C6;&#x2019;1 u i = ui+1 , vi = vi+1 , u i+1 = ui TÎ¨,i+1 , vi+1 = TÎ¨,i+1 vi .

Remarks 3.4. (a) Note that perverse sheaves being a topological concept, the group Diff + (â&#x2C6;&#x2020;; B, b) naturally acts on the category Perv(â&#x2C6;&#x2020;, B) from the first principles, the action descending to that of Brn . (b) We can turn Proposition 3.1 and formulas (7) around to produce an intrinsic (i.e., not tied to any particular K and manifestly Brn equivariant) definition of Perv(â&#x2C6;&#x2020;, B) which does not appeal to any preexisting concept of a perverse sheaf. More precisely, we can define an object P â&#x2C6;&#x2C6; Perv(â&#x2C6;&#x2020;, B) to be a system of objects PK â&#x2C6;&#x2C6; Pn , K â&#x2C6;&#x2C6; C and compatible isomorphisms fĎ&#x192; (PK ) â&#x2020;&#x2019; PĎ&#x192;(K) , Ď&#x192; â&#x2C6;&#x2C6; Brn so that each particular PK is just a particular â&#x20AC;&#x153;shadowâ&#x20AC;? of a more intrinsic object P . B. Schobers on a disk with several marked points. To give an â&#x20AC;&#x153;invariant" definition of a perverse Schober on (â&#x2C6;&#x2020;, B), we adopt the approach of Remark 3.4(b). That is, for each system of cuts K â&#x2C6;&#x2C6; C we define a K-coordinatized Schober to be a system of n spherical functors with a common target SK = Si : Di â&#x2C6;&#x2019;â&#x2020;&#x2019; D, i = 1, Âˇ Âˇ Âˇ , n .

Mikhail Kapranov and Vadim Schechtman

Each SK gives rise to spherical reflection functors Ti = Cone{Si Ri ⇒ IdD }, i = 1, · · · , n. According to our convention on working with triangulated categories in terms of dg-enhancements, see Appendix, all K-coordinatized Schobers form an ∞-category which we denote SchK (∆, B). We define equivalences fσ : SchK (∆, B) → Schσ(K) (∆, B), σ ∈ Brn on generators si ∈ Brn by the direct analog of (7): fsi Si : Di −→ D = Si : Di −→ D , (8)

D = D, Dj = Dj , Sj = Sj , = Di , Di Di+1 Si = Si+1 , Si+1

j = i, i + 1,

= Di+1 ,

−1 = Ti+1 Si .

We then extend to arbitrary σ ∈ Brn by verifying the braid relations for the fi which is done in exactly the same way as for (7). By definition, a perverse Schober on (∆,B) is a system S = (SK )K∈C of coordinatized Schobers and compatible identifications fσ (SK ) → Sσ(K) . The datum SK will be referred to as the K-shadow of S. We denote by Sch(∆, B) the ∞-category of perverse Schobers on (∆, B). C. The Picard-Lefschetz formula. Underlying classical Picard-Lefschetz theory, there is a general statement about perverse sheaves on a disk which we now formulate. Let F ∈ Perv(∆, B). In the approach of (5), “the” space of vanishing cycles of F at some bi ∈ B can be defined in terms of a small segment of an arc terminating in bi (which does not, a priori, have to be a part of a system of cuts). Let now γ be a simple arc joining two marked points bi and bk and not passing through any other marked points, as in Figure 3. We can then define the spaces of vanishing and nearby cycles of F relative to γ: Φi,γ = H1γ (F)bi , Φk,γ = H1γ (F)bk , Ψγ = H1γ (F)gen (generic stalk). which are connected by the maps Φi,γ

vi,γ ui,γ

Ψ k,γ Φ . γ u k,γ k,γ

Perverse Schobers

The definition of these maps is similar to §2A: the maps v are generalization maps, and the maps u are obtained by counterclockwise continuation, as in (3). We define the transition map along γ as Mik (γ) = uk,γ ◦ vi,γ : Φi,γ −→ Φk,γ . The data of the Φi (F), i = 1, · · · , n as local systems on the circles around bi and of all the Mik (γ) describe the image of F in the localization of Perv(∆, B) by the subcategory of constant sheaves [21, §2].

∆ bi β bj

K γ

α bk

Figure 3: The Picard-Lefschetz situation. One can say that “abstract Picard-Lefschetz theory” is the study of how Mik (γ) changes when we replace γ by a different (non-isotopic) arc γ . More precisely, assume that γ is obtained from γ by an “elementary move” past another marked point bj so that the bigon formed by γ and γ contains bj and two arcs α and β, but no other marked points as in Fig. 3. In particular, we assume that the closed path obtained by following γ from bi to bk and then γ from bk to bi , has orientation compatible with the standard (counterclockwise) orientation of ∆. Note that homotopy inside the bigon and the clockwise rotation around bj give identifications (9)

Φi,γ Φi,β Φi,γ ,

Φk,γ Φk,α Φk,γ ,

Φj,β Φj,α ,

so we can consider them as single spaces denoted by Φi , Φk and Φj respectively.

Mikhail Kapranov and Vadim Schechtman

Proposition 3.5 (Picard-Lefschetz formula for perverse sheaves). We have the equality of linear operators Φi → Φk : Mik (γ ) = Mik (γ) + Mjk (α)Mij (β).

This statement is a version of [21, Prop. 2.4], formulated in a more invariant way and without localizing by constant sheaves. It holds for perverse sheaves on any oriented surface. It is convenient to give two proofs of Proposition 3.5. Invariant proof: To eliminate the need for the first two identifications in (9), let us deform the paths α, β, γ, γ so that: • γ, β, γ have a common segment [bi , b i ] near bi . • γ, α, γ have a common segment [b k , bk ] near bk . See Fig. 4.

We denote by γ, γ the parts of γ and γ lying between b i and b k , by β the part of β between b i and bj , and by α the part of α between bk and b j . bi

γ b k

b i γ

γ β

Figure 4: The Picard-Lefschetz situation, deformed. The points b i , b k being smooth for F, both sides of our putative equality factor through the maps v[bi ,b i ] : Φi −→ Fb i ,

u[b k ,bk ] : Fb i −→ Φk .

Our statement would therefore follow from the next lemma which is an invariant version of the statement that the map (2) is indeed the monodromy around 0. Lemma 3.6. We have an equality of operators Fb i → Fb k : Tγ = Tγ + vα Ruβ , where:

Perverse Schobers

(1) uβ : Fb i → H1β (F)bj is the coboundary of the counterclockwise continuation map, cf. (3). (2) R : H1β (F)bj → H1α (F)bj is the identification obtained by deform-

ing, by clockwise rotation aroung bj , the path β into the path α, i.e., R is the third identification in (9).

(3) vα : H1α (F)bj → H1α (F)gen Fb k is the generalization map of the sheaf H1α (F). Proof of the lemma: Let U be a small disk around bj . We fix a ∈ Fb and compare the two sides of the putative equality when applied to a. For this, let sa ∈ Γ(U − β, F) be the section obtained by continuing a on the left side of β towards and around bj . Then uβ (a) is equal to the class of sa in H1β (F)bj . Next, Ruβ (a) is similarly represented by the section ta obtained from sa by continuously moving the branch cut clockwise from α to β. This means that ta = sa on the left side of α ∪ β. Finally, vα Ruβ (a) is obtained as the difference of the two boundary values if ta when continued along both sides of α all the w ay to b k . It remains to notice that these boundary values are equal, in virtue of the above, to . Tγ (a) and Tγ (a). Proof using shadows: Choose a system of cuts K (depicted by dotted lines in Fig. 3) adopted to our situation. We assume that the arcs Ki , Kj and Kk are positioned as in the figure, i.e., that γ together with Ki and Kk form a triangle containing Kj , α and β and not containing any other marked points. We orient each Kν to run from b to bν . Consider the quiver FK (F) = (Ψ, Φi , ui , vi ). Note that we have isotopies of oriented paths rel. B: α ∼ Kk ∗ Kj−1 , β ∼ Kj ∗ Ki−1 , γ ∼ Kk ∗ Ki−1 , γ ∼ Kk ∗ ∂∆ ∗ Ki−1 .

Here ∗ means composition of the paths and ∂∆ is the boundary circle of ∆, oriented anticlockwise and run from b to b. We can use these isotopies to calculate the transition maps, obtaining Mij (β) = uj,K vi,K ,

Mjk (α) = uk,K vj,K ,

Mik (γ ) = uk,K vi,K ,

Mik (γ) = uk,K Tj,Ψ vi,k , and the claim follows from the identity Tj,Ψ = Id − vi uj .

100

Mikhail Kapranov and Vadim Schechtman

D. The Picard-Lefschetz triangle. Let S be a perverse Schober on (∆, B). The transition maps constructed in n◦ C categorify to functors between triangulated categories. More precisely, if γ is an oriented arc joining bi and bk as above, then we have a diagram of triangulated categories and spherical functors which depends “canonically" (i.e., up to a contractible set of choices) only on the isotopy class of γ rel. B: (10)

Φi,γ (S)

Si,γ

Ψγ (S)

Sk,γ

Φk,γ (S).

To define it, we choose K ∈ C so that Ki (oriented from b to bi ), together with γ and (Kk )−1 form a positively oriented triangle not containing any other marked points. (Note: the choice K as in Fig.3 is not good.) of n We consider the K -shadow SK = SνK : DνK → DK ν=1 of S and define Ψγ (S) = DK , Φi,γ (S) = DiK , Φk,γ (S) = DkK ,

Si,γ = SiK , Sk,γ = SkK .

It is straightforward to see (by looking at the subgroup in Brn permuting all K with our property) that this definition is indeed “canonical” in the sense described. So we consider the data (10) as intrinsically associated to S and γ. Denote Rk,γ the right adjoint to the spherical functor Sk,γ . Define now the transition functor Mik (γ) = Rk,γ ◦ Si,γ : Φi,γ (S) −→ Φk,γ (S). Consider now a situation when we have arcs γ, γ , α, β depicted in Fig. 3. Using identifications of categories similar to (9), we can speak about triangulated categories Φi (S) and Φk (S). Proposition 3.7 (Picard-Lefschetz triangle). We have a canonical triangle of exact functors Φi (S) → Φk (S): Mik (γ) −→ Mik (γ ) −→ Mjk (α) ◦ Mij (β) −→ Mik (γ)[1]. Proof: This is obtained identically to the “shadow" proof of Proposition 3.5 with the identity Tj,Ψ = Id − vj uj replaced by the triangle coming from the fact that Tj = Cone{Id → Rj Sj }. E. Schobers on a Riemann surface. Let Σ be an oriented topological surface, possibly with boundary ∂Σ, and B ⊂ Σ is a finite set not meeting ∂Σ. One can then define a perverse Schober on (Σ, B) by

Perverse Schobers

101

decomposing Σ as ∆ ∪C U , where ∆ ⊂ Σ is a closed disk with boundary circle C = ∂∆ which contains all points of B, and U is the closure of Σ − ∆. Then a perverse Schober S is, by definition, a datum of: (1) A perverse Schober S∆ on (∆, B), defined as in n◦ B. (2) A local system of triangulated categories SU on U identified with S∆ over C. While one can work with objects thus defined (for instance, one can construct transition functors and Picard-Lefschet triangles for arcs not necessarily contained in ∆), a more intrinsic definition is desirable.

102

Mikhail Kapranov and Vadim Schechtman

Spherical functors and spherical pairs

A. Symmetric description of Perv(∆, 0). In [24, §9] we have given a different quiver description of Perv(∆, 0) obtained as a particular case of a general result for real hyperplane arrangements: Proposition 4.1. The category Perv(∆, 0) is equivalent to the category formed by diagrams of vector spaces (11)

E−

γ− δ−

γ+

δ+

E+

satisfying the two following conditions: (1) γ− δ− = IdE− , γ+ δ+ = IdE+ . (2) The maps γ− δ+ : E+ → E− , γ+ δ− : E− → E+ are invertible.

This is obtained by choosing not one but two base points b+ , b− ∈ ∂∆ and considering a cut K which joins b+ with b− and passes through 0, as in depicted in Fig. 5. ∆

b−

Figure 5: A symmetric cut. The spaces E± , E0 are obtained as the stalks of the sheaf H1K (F) at b± and 0 respectively, the maps γ± are the generalization maps for this sheaf, and the δ± can be obtained by duality. Remarks 4.2. (a) Note that in this description the maps P+ = δ− γ− and P+ = δ+ γ+ are projectors in E0 , that is P±2 = P± . We can consider E± as subspaces in E0 which are the images of P± .

(b) The pattern of Remarks 2.3 and 3.2 continues here: HiK (F) = 0 for i = 1 and each stalk of H1K considered as a functor into Vectk , takes Verdier duality into vector space duality.

Perverse Schobers

103

B. Reminder on semi-orthogonal decompositions. A categorification of diagrams (11) is naturally formulated in the language of semiorthogonal decompositions of triangulated categories [10] [27] which we now recall. Let B be a full triangulated subcategory of a triangulated category A, with i = iB : B → A being the embedding functor. We denote by ⊥

B = {A ∈ A| Hom(A, B) = 0 ∀B ∈ B},

B ⊥ = {A ∈ A| Hom(B, A) = 0 ∀B ∈ B}

the left and right orthogonals to B We say that B is left admissible, resp. right admissible, if i has a left adjoint ∗ i, resp. a right adjoint i∗ . If B is left (resp. right) admissible, we have a semi-orthogonal decomposition A = ⊥ B, B ,

resp. A = B, B ⊥

which means that each object A ∈ A is included in functorial exact triangles C −→ A −→ B → C[1],

resp. B −→ A −→ D → B [1],

C ∈⊥ B, B = ∗ i(A) ∈ B,

B = i∗ (A) ∈ B, D ∈ B ⊥ .

In particular, ⊥ B = Ker(∗ i), resp. B ⊥ = Ker(i∗ ). We will call ∗ i the projection on B along ⊥ B and i∗ the projection on B along B ⊥ . If B is left admissible, then ⊥ B is right admissible, and (⊥ B)⊥ = B. Similarly, for a right admissible B we have that B ⊥ is left admissible and ⊥ (B ⊥ ) = B. We say call B admissible, if it is both left and right admissible. For an admissible B we have an equivalence MB : B ⊥ −→ ⊥ B, known as the mutation along B, see [10]. It is defined as the composition i

⊥

i∗⊥

B ⊥ B A −→ B B ⊥ −→

of the embedding of B ⊥ and of the projection onto ⊥ B along B. Proposition 4.3. If B is admissible, then the functor ∗ iB⊥ (the projection onto B ⊥ along B) has itself a left adjoint ∗∗ iB⊥ : B ⊥ → A given by ∗∗ i B ⊥ = i⊥ B ◦ M B .

104

Mikhail Kapranov and Vadim Schechtman

Proof: Let B ∈ B ⊥ and A ∈ A. The definitions of MB and ∗ iB⊥ give exact triangles i⊥ B (MB (B )) −→ B −→ B −→ i⊥ B (MB (B ))[1], B1 −→ A −→ ∗ iB⊥ (A) −→ B1 [1]

with B, B1 ∈ B. These triangles give canonical identifications (1)

(2)

HomA (i⊥ B (MB (B )), A) HomA (i⊥ B (MB (B )), ∗ iB⊥ (A)) (2)

(3)

HomA (B , ∗ iB⊥ (A)) = HomB⊥ (B , ∗ iB⊥ (A))

with the reasons being: (1) since Hom•A (i⊥ B (MB (B )), B1 ) = 0; (2) since Hom•A (B, ∗ iB⊥ (A)) = 0; (3) since B ⊥ is a full subcategory in A. Combined together, the identifications (1)-(3) give the claimed adjointness. C. Spherical pairs. Let E0 be a triangulated category and E+ , E− ⊂ E0 be a pair of admissible subcategories, so that we have the diagrams of embeddings δ−

δ+

E− −→ E0 ←− E+ ,

j−

⊥ ⊥ E− −→ E0 ←− E+

∗ , ∗ δ and j having a left with δ± having a left and a right adjoint δ± ± ± and a double left adjoint ∗ j± , ∗∗ j± by Proposition 4.3.

Definition 4.4. The pair of admissible subcategories E± is called a spherical pair, if: (SP1) The compositions ∗

⊥ ⊥ j+ ◦ j − : E− −→ E+ ,

∗

⊥ ⊥ j− ◦ ∗j+ : E+ −→ E−

are equivalences. (SP2) The compositions ∗ δ+ ◦ δ− : E− −→ E+ ,

are equivalences.

∗ δ− ◦ δ+ : E+ −→ E−

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Remark 4.5. Note that unlike a spherical functor, the definition of a spherical pair does not appeal to any enhancement of the triangulated category E0 , as no functorial cones are taken. In the following (as elsewhere in the paper) we will, however, assume that we are in an enhanced situation as described in Appendix. D. From a spherical pair to a spherical functor. Let E± be a spherical pair. Consider the diagram S

⊥ = D1 , D0 = E− −→ E+

S = ∗ j + ◦ δ− .

Proposition 4.6. S is a spherical functor. This follows from Theorem 2.5 (which gives that (SF1) and (SF3) imply sphericity) and from the next more precise statement. Proposition 4.7.

(a) The functor S has both right and left adjoints

∗ ◦ j+ , R = S ∗ = δ−

L = ∗ S = ∗ δ− ◦ ∗∗ j+ .

(b) The functor ⊥ ⊥ T1 = Cone{SR ⇒ IdD1 } : D1 = E+ −→ E+ = D1 ⊥ → E ⊥. is identified with the composition ∗ j+ ◦ j− ◦ ∗ j− ◦ j+ : E+ + In particular, it is invertible.

(c) The functor T0 = Cone{IdD0 ⇒ RS}[−1] : D0 = E− −→ E− = D0 ∗ ◦ δ ◦ δ ∗ ◦ δ : E → E . In is identified with the composition δ− + − − − + particular, it is invertible.

Proof: (a) obvious from the assumptions and the fact that the adjoint of the composition of two functors is the composition of the adjoints in the opposite order. (b) The functorial exact triangle for the semi-orthogonal decompo⊥ can be written, in our notation, as sition E0 = E− , E− u−

v−

w−

∗ ∗ =⇒ IdE0 =⇒ j− ◦ ∗ j− =⇒ δ− ◦ δ− [1]. δ − ◦ δ−

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â&#x2C6;&#x2014; â&#x2014;¦ j and the natural transformation We note that SR = 8 j+ â&#x2014;¦ Î´â&#x2C6;&#x2019; â&#x2014;¦ Î´â&#x2C6;&#x2019; + c : SR â&#x2021;&#x2019; IdD1 can be written as â&#x2C6;&#x2014;

â&#x2C6;&#x2014;

j + â&#x2014;¦0 u â&#x2C6;&#x2019; â&#x2014;¦0 j + :

â&#x2C6;&#x2014; â&#x2014;¦ j+ =â&#x2021;&#x2019; â&#x2C6;&#x2014; j+ â&#x2014;¦ j+ = IdD1 . j + â&#x2014;¦ Î´ â&#x2C6;&#x2019; â&#x2014;¦ Î´â&#x2C6;&#x2019;

Here â&#x2014;¦0 stands for the â&#x20AC;&#x153;0-composition" of a functor and a natural transformation. This implies that Cone(c) =

â&#x2C6;&#x2014;

j+ â&#x2014;¦ j â&#x2C6;&#x2019; â&#x2014;¦ â&#x2C6;&#x2014; jâ&#x2C6;&#x2019; â&#x2014;¦ j +

as claimed. â&#x160;¥ as (c) We now write the functorial exact triangle for E0 = E+ , E+ u+

â&#x2C6;&#x2014; â&#x2C6;&#x2014; =â&#x2021;&#x2019; IdE0 =â&#x2021;&#x2019; j+ â&#x2014;¦ â&#x2C6;&#x2014; j+ =â&#x2021;&#x2019; Î´+ â&#x2014;¦ Î´+ [1]. Î´ + â&#x2014;¦ Î´+ â&#x2C6;&#x2014; â&#x2014;¦ j â&#x2014;¦ â&#x2C6;&#x2014; j â&#x2014;¦ Î´ and the natural transformation We note that RS = Î´â&#x2C6;&#x2019; + + â&#x2C6;&#x2019; : IdD1 â&#x2021;&#x2019; RS can be written as â&#x2C6;&#x2014; â&#x2C6;&#x2014; â&#x2C6;&#x2014; Î´â&#x2C6;&#x2019; â&#x2014;¦0 v+ â&#x2014;¦0 Î´â&#x2C6;&#x2019; : IdD0 = Î´â&#x2C6;&#x2019; â&#x2014;¦ Î´â&#x2C6;&#x2019; =â&#x2021;&#x2019; Î´â&#x2C6;&#x2019; â&#x2014;¦ j + â&#x2014;¦ â&#x2C6;&#x2014; j+ â&#x2014;¦ Î´â&#x2C6;&#x2019; .

This implies that â&#x2C6;&#x2014; â&#x2C6;&#x2014; Cone( )[â&#x2C6;&#x2019;1] = Î´â&#x2C6;&#x2019; â&#x2014;¦ Î´ + â&#x2014;¦ Î´+ â&#x2014;¦ Î´â&#x2C6;&#x2019;

as claimed. Exercise 4.8 (Polar coordinates). Let Y be a CW-complex and p : V â&#x2020;&#x2019; Y a real vector bundle. Denote by i : Y â&#x2020;&#x2019; V the embedding of the zero section, and by j : V â&#x2014;¦ â&#x2020;&#x2019; V the embedding of the complement of the zero section. Let q : S = V â&#x2014;¦ /Râ&#x2C6;&#x2014;>0 â&#x2020;&#x2019; Y be the spherical bundle associated to V , and Ï&#x201E; : V â&#x2014;¦ â&#x2020;&#x2019; S the natural projection. Let also Ï&#x20AC; = qÏ&#x201E; : V â&#x2014;¦ â&#x2020;&#x2019; Y be the composite projection. b (V ) â&#x160;&#x201A; Db (V ) be the full subcategory of Râ&#x2C6;&#x2014;>0 Let E0 = E0 (V ) = Dconic conic complexes, i.e., of complexes F such that each H i (F) is constant on each orbit of Râ&#x2C6;&#x2014;>0 in V , see [25]. Consider the subcategories EÂ± = EÂ± (V ) â&#x160;&#x201A; E0 defined as follows: E+ = iâ&#x2C6;&#x2014; Db (Y ) Db (Y ), Then and the functor

Eâ&#x2C6;&#x2019; = pâ&#x2C6;&#x2019;1 Db (Y ) Db (Y ).

â&#x160;¥ E+ = Rjâ&#x2C6;&#x2014; Ï&#x201E; â&#x2C6;&#x2019;1 Db (S) Db (S), â&#x160;¥ S = â&#x2C6;&#x2014; j+ â&#x2014;¦ Î´â&#x2C6;&#x2019; : Eâ&#x2C6;&#x2019; â&#x2C6;&#x2019;â&#x2020;&#x2019; E+

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is identified with the spherical functor q −1 from Example 2.6. This is an instance of the above proposition, since (E± ) form a spherical pair. To see this, it suffices to show that both E+ and E− are admissible (the conditions (SP1-2) are proved similarly to the fact that S is spherical). For E+ this is clear from the standard “recollement” data for complexes of sheaves on the open set V ◦ and the closed set i(Y ). In particular, ⊥

E+ = Rj! τ −1 Db (S).

For E− this follows from the next remark. Remark 4.9. In the situation of Example 4.8, let V ∗ be the vector bundle dual to V , so that the Fourier-Sato transform [25] gives an identification F

b b E0 (V ) = Dconic (V ) −→ Dconic (V ∗ ) = E0 (V ∗ ).

This identification takes the category E± (V ) to E∓ (V ∗ ).

Derived categories on the G/P and Schobers on symmetric products h/W .

A. The braid group action on Db (G/B) and Dcoh (T ∗ (G/B)). Let g be a split reductive Lie algebra over C, with Cartan subalgebra h and Weyl group W . We denote by hR ⊂ gR the real parts of h and g. Let ∆sim ⊂ ∆+ ⊂ ∆ be the systems of simple and positive roots of g inside the set of all roots. The complex vector space h has an arrangement of hyperplanes {α⊥ }α∈∆+ and so has a natural stratification by flats of this arrangements, see [24]. We consider the complex manifold h/W . It has the induced stratification, denote it S. The open stratum (h/W )0 , is the classifying space of the braid group of g, which we denote by Br = Br(g). By definition, Br is generated by elements sα , α ∈ ∆sim subject to the relations defining the Weyl group W with the exception of the relations s2α = 1. Exercise 5.1. In the case g = gln , the manifold h/W is space of monic polynomials f (z) of degree n in one variable. The stratification S is the stratification by types of coincidence of roots of f (z) and is labelled by (unordered) partitions of n. The group Br(gln ) is the usual Artin braid group Brn .

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Let B ⊂ G be the standard Borel subgroup corresponding to the choice of ∆+ . For any I ⊂ ∆sim let PI ⊃ B be the standard parabolic subgroup generated by B and the exponents of the Chevalley generators associated to (−α), α ∈ / I. Thus P∅ = G and P∆sim = B. We write Pα = P∆sim −{α} , α ∈ ∆sim , for the “next to minimal" parabolic subgroup associated to {α}. We have the P1 -fibration qα

G/B −→ G/Pα and therefore, by Proposition 2.7, a spherical functor −1 Sα =qα

Db (G/Pα ) −→ Db (G/B) and the corresponding twist functor Tα : Db (G/B) → Db (G/B) written directly as (12) Tα (F) = R(pα,2 )! p∗1,α F[−1], pα,i : (G/B) ×G/Pα (G/B) − rel. diagonal −→ G/B, i = 1, 2. The following fact was known before the concept of a spherical functor was discovered, cf. [8].

Proposition 5.2. The functors Tα defined by (12), are equivalences which satisfy the relations of Br(g), i.e., define an action of Br(g) on the derived category Db (G/B). Remarks 5.3. (a) Proposition 5.2 for g = gl(n) has a quaternionic analog. We consider the space HFn of complete flags of (left) quaternionic subspaces V1 ⊂ · · · ⊂ Vn−1 ⊂ Hn , dimH (Vi ) = i. It fits into the HP1 -fibrations qi : HFn −→ HFn(i) = (V1 ⊂ · · · ⊂ Vi−1 ⊂ Vi+1 ⊂ · · · ⊂ Hn over the spaces of next-to-complete flags. We note that HP1 = S4 is the 4-sphere. By Proposition 2.7 this gives, for each i = 1, · · · , n − 1, (i) a spherical functor SiH = qi−1 : Db (HFn ) → Db (HFn ) and the induced twist automorphism TiH of Db (HFn ). One can see directly by analyzing the Schubert correspondences in HFn , that the TiH satisfy the relations of the Artin braid group Brn . (b) One has also a real analog for any g, using the real loci of the flag varieties which fit into RP1 = S1 -fibrations.

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A quasi-classical analog of the action in Proposition 5.2 has been constructed in [26, 9]. It uses the diagram ρα

α Yα := T ∗ (G/Pα ) ←− Dα := T ∗ (G/Pα ) ×G/Pα G/B −→ T ∗ (G/B) := X

which produces a spherical functor b b T ∗ (G/Pα ) → Dcoh T ∗ (G/B) Sαcoh = iα∗ ρ∗α : Dcoh

It was proved in [26] for g = gln and in [9] for arbitrary g, that the correb T ∗ (G/B) sponding twist functors Tαcoh define an action of Br(g) on Dcoh (which, in fact, extends to an action of the affine braid group). These constructions can be seen as giving local systems of triangulated categories on (h/W )0 = K(Br(g), 1) with general stalk being b (T ∗ (G/B)). We would like to suggest that these Db (G/B), resp. Dcoh local systems extend to natural perverse Schobers on the entire h/W . For this, we review some features of usual perverse sheaves in this situation. B. Perverse sheaves on h/W and double cubical diagrams. Denote Perv(h/W ) the category of perverse sheaves on h/W smooth with respect to the stratification S from n◦ A. A complete quiver description on Perv(h/W ) is not yet available. However, the results of [24] provide the following partial picture which aligns with the examples we considered earlier. Let hR ⊂ h be the real form of h. Inside h/W we consider the “real skeleton" (or “cut”) K = hR /W ⊂ h/W.

It can be thought of as a “curvilinear cone", the image of the dominant sim | be the rank of g. Let S be the Weyl chamber h+ R R ⊂ hR . Let r = |∆ stratification of K by the 2r strata which are the images of the faces of sim , so that dim (S ) = |I|. h+ R I R . We denote these strata by SI , I ⊂ ∆ Note that each SI is contractible. Therefore (cf. [20]) a sheaf G on K constructible with respect to SR can be recovered from the cubical diagram of vector spaces GI = Γ(SI , G) and generalization maps γII : GI → GI , I ⊂ I which are required to form a representation of the sim poset 2∆ (so that the diagram is commutative). The following is deduced by pulling F back to a perverse sheaf on h smooth with respect to an arrangement of hyperplanes {α⊥ }α∈∆+ and applying the results of [24].

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Proposition 5.4. (a) For F ∈ Perv(h/W ) we have HiK (F) = 0 for dim(K) (F) on K is constructible i = dimR (K), and the sheaf RF = HK with respect to the stratification SR . (b) Let I ⊂ ∆sim . Denoting EI (F) = Γ(SI , RF ), we have that EI : Perv(h/W ) → Vectk is an exact functor which takes the Verdier duality to the vector space duality. Therefore we can associate to any F ∈ Perv(h/W ) a double cubical diagram E(F) formed by the vector spaces EI (F) and the maps (13)

EI (F)

γIJ

EJ (F) ,

δJI

I ⊂ J.

Here γIJ is the generalization map for EF , and δJI is the dual to the generalization map for EF ∗ , where F ∗ is the perverse sheaf Verdier dual to F. Each of the collections (γIJ ), (δJI ) forms a commutative cube. It seems very plausible that the functor F → E(F) from Perv(h/W ) to the category of double cubical diagrams is fully faithful, i.e., F can be recovered from E(F). Exercise 5.5. For g = sl2 we have h/W = C and K = R≥0 so the above reduces, very precisely, to the construction of §2A, except with the disk replaced by C. C. A double cubic diagram related to flag varieties. We now note that geometry of flag varieties provides a natural double cubic diagram of categories of the same shape as (13). More precisely, we have an ordinary cubical diagram of algebraic varieties G/PI and projections qIJ : G/PJ → G/PI , I ⊂ J. We have then the double cubical diagram formed by triangulated categories Db (G/PI ) and the adjoint pairs of functors (14)

Db (G/P

−1 qIJ

Db (G/PJ ) ,

RqIJ∗

I ⊂ J.

−1 which As in Example 2.6, we can also consider the left adjoint to qIJ differs from the right adjoint by a shift. At the “quasi-classical level” one has a similar diagram formed by the b (T ∗ (G/P )) and the functors between them obtained by categories Dcoh I −1 translating qIJ and RqIJ∗ into the language of D-modules and passing to the associated graded modules.

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We would like to suggest that the diagram (14) comes from a more fundamental object: a perverse Schober on h/W extending the local system discussed in n◦ A. b (T ∗ (G/P )). In Similarly for the quasi-classical analog with the Dcoh I this situation we have in fact more: an actions the affine braid group. These can possibly come from Schobers not on h/W but on T /W where T is the maximal torus in the algebraic group corresponding to g.

“Fukaya-style” approach to perverse sheaves: cuts, real skeletons and Langangian varieties

A. Maximally real cuts. Since our preliminary definitions of a perverse Schobers were based on quiver descriptions of perverse sheaves, let us look at some general features of such descriptions. Let (X, S) be a stratified complex manifold of dimension n. Obtaining a quiver description of Perv(X, S) requires, in particular, construction of many exact functors Perv(X, S) → Vectk . Indeed, any component of the putative quiver must be such a functor. Arrows of the quiver are then natural transformations between these exact functors. The common tool for that, used in examples in this paper, is a choice of a closed subset K ⊂ X with the following property: (Cut) For any F ∈ Perv(X, S) we have HiK (F) = 0, i = n. This property implies that the functor of abelian categories R : F −→ R(F) := HnK (F),

Perv(X, S) −→ ShK

is exact. So stalk of R(F) at any point x ∈ K can be used as a component of a quiver, while generalization maps between the stalks provide some of the arrows. We will call each K satisfying (Cut) an (admissible) cut for (X, S) and denote by C(X, S) the set of all such cuts. It is natural to look for a description of Perv(X, S) in terms of some data associated to all the cuts. Recall that by the Riemann-Hilbert correspondence we can realize each F ∈ Perv(X, S) as R HomDX (M, OX ) for a left DX -module M whose characteristic variety satisfies the inclusion TX∗ α X. Ch(M) ⊂ ΛS := α

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We denote by D Mod(X, S) the category of such DX -modules. In these terms, a more detailed scenario (sufficient condition) for (Cut) to hold would be for K to satisfy the following two properties: (Cut1) We have HiK (OX ) = 0 for i = n. (Cut2) Assuming (Cut1), the sheaf BK = HnK (OX ) (which, considered as a sheaf on X, is automatically a sheaf of left DX -modules), satisfies ExtjDX (M, BK ) = 0,

∀M ∈

Mod(X, S), j > 0.

The property (Cut1) is not related to a choice of S and holds for any totally real subset of X, see [22]. More precisely, we recall, see [5] for background: Definition 6.1. (a) Let V be a C-vector space of finite dimension n. A real subspace L ⊂ V is called totally real, if L ∩ iL = 0. We say that L is maximally real, if it is totally real of dimension n. (b) Let X be a complex manifold of dimension n. A C ∞ -submanifold K ⊂ X is called totally real, resp. maximally real if for each x ∈ K the subspace Tx K ⊂ Tx X is totally real resp. maximally real. The results of Harvey [22] imply:

Proposition 6.2. Any closed subset of a totally real submanifold of X satisfies (Cut1). More precisely, the result of loc. cit. is for arbitrary totally real subsets of X, a class of sets which includes totally real submanifolds [22, §3.6 Ex. 2] and is closed under passing to closed subsets [22, Cor. 3.2]. For example, Rn , as well as Rn≥0 satisfies (Cut1). The sheaf BRn is the sheaf of hyperfunctions of Sato [32]. We now consider the condition (Cut2). Sufficient criteria for it to hold were given by Lebeau [28] and Honda-Schapira [23]. The criterion of [23] is based on the concept of positive position of two real Lagrangian submanifolds in a complex symplectic manifold such as T ∗ X. We do not recall this concept here, referring to [33] and references therein for more background. Informally, the essense of the criterion can be formulated like this. For (Cut2) to hold, K must be “maximally real with respect to (15)

the stratification S”, in particular, the intersection of K with each stratum Xα should be maximally real in Xα .

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Here is a precise but more restrictive statement which is a reformulation of Example 1 of [23]. Proposition 6.3. Suppose S is such that ΛS is contained in the union of TZ∗β X for a collection of smooth closed complex submanifolds Zβ ⊂ X. Suppose K ⊂ X is a maximally real analytic submanifold such that each K ∩ Zβ is maximally real in Zβ . Then K satisfies (Cut2). B. Maximally real vs. Lagrangian cuts. Let V be a complex vector space of dimension n and GR (n, V ) the Grassmannian of real ndimensional subspaces in V . We denote by Gmax (V ) the open subset in GR (n, V ) formed by maximally real subspaces. Suppose V is equipped with a positive definite hermitian form h. Separating the real and imaginary parts h = g + iω, we have that ω is a symplectic form on V . Let LGω (V ) be the closed subset in GR (n, V ) formed by subspaces Lagrangian with respect to ω. The following is well known. Proposition 6.4. LGω (V ) is contained in Gmax (V ) and the embedding is a homotopy equivalence. In other words, LGω (V ) can be seen as a compact form of Gmax (V ). Proof: For V = Cn with the standard hermitian form we have LGω (V ) = O(2n)/U (n) ⊂ GL(2n, R)/GL(n, C) = Gmax (V ). Let now X be a complex manifold equipped with a Kähler metric h = g + iω. Then ω makes X into a symplectic manifold, and we have Corollary 6.5. Any Lagrangian submanifold of X is maximally real. Note that all the cuts used in this paper as well as in [19] [24], are Lagrangian. This suggests a possibility of describing more general Perv(X, S) in terms of data coming from Lagrangian cuts.

On the other hand, Lagrangian submanifolds of a Kähler manifold X are organized into a far-reaching structure: the Fukaya category F(X) (as well as its modifications such as the Fukaya-Seidel and wrapped Fukaya categories), see [18] [34] for background. We briefly recall that F(X) is an (a priori partially defined) C-linear A∞ category whose objects are,

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in the simplest setting, compact Lagrangian submanifolds K ⊂ X. The space Hom(K1 , K2 ) is defined when K1 and K2 meet transversely and in this case is formally spanned by the set K1 ∩ K2 (with appropriate grading, see loc. cit.). The A∞ -composition µK1 ,··· ,Km :

m−1 i=1

Hom(Ki , Ki+1 ) −→ Hom(K1 , Km ),

m ≥ 1,

is given by counting holomorphic3 disks D ⊂ X with boundary on K1 ∪ · · · ∪ Km . Each D gives a contribution exp − D ω to the appropriate matrix element of µK1 ,··· ,Km .

One can expect that the Fukaya structure on the collection of admissible Lagrangian cuts for (X, S) has some significance for explicit description of Perv(X, S). In particular, a holomorphic disk D with boundary on the union of admissible cuts K1 , · · · , Km may provide a link between the sheaves HnKν (F), F ∈ Perv(X, S) for different ν, via the structure of perverse sheaves on a disk (§3).

Remarks 6.6. (a) Any collection K1 , · · · , Km of maximally real (not necessarily Lagrangian) submanifolds in X provides a natural boundary condition for holomorphic disks. It has been noticed [34] that many ingredients of the Fukaya category construction have an “intrinsic" meaning and can be defined without explciit reference to the symplectic structure. In particular: • The role of the grading on Hom(K1 , K2 ) is to make the count of disks possible by ensuring that in evaluating matrix elements of µ between basis vectors of the same degree, the index of the linearlized elliptic problem is 0 (so the corresponding linear ∂-operator is, generically, invertible). • The quantity exp − D ω can be intepreted as the determinant of the invertible ∂-operator above. They could therefore make sense for more general maximally real submanifolds. The usual technical reason for restricting to Lagrangian submanifolds in defining F(X) is that the Gromov compactness theorem (which, via the properties of 1-dimensional moduli spaces, is used to 3 Here we leave aside the additional complication that it may be necessary to deform the complex structure on X to ensure generic position.

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115

prove the A∞ -axioms) has originally been established only in that setting. However, see [17] for a recent generalization to the maximally real case. (b) the above mentioned role of cuts (in particular, Lagrangian cuts) for description of perverse sheaves on X, is different from the classical “microlocal” point of view [25] which empasizes complex, conic Lagrangian subvarieties in T ∗ X rather than real ones in X itself. For the relation of the microlocal approach to the Fukaya category (of T ∗ X), see [31][30]. A more general idea that the Fukaya category of any symplectic manifold should have an interpretation in terms of its geometric quantization, was proposed earlier in [12]. C. Perverse Schobers as coefficients data for Fukaya categories. The Fukaya category F(X) of a symplectic manifold X can be seen as a categorification of its middle (co)homology (or, rather, of the part represented by Lagrangian cycles). Here “cohomology” is understood as H • (X, C), the cohomology with constant coefficients, a particular case of a more general concept of H • (X, F), the cohomology with coefficients in a sheaf F or even more generally, in a complex of sheaves. This point of view leads to the idea of introducing coefficients into the definition of the Fukaya category as well. It was proposed by M. Kontsevich with the goal of understanding the (usual) Fukaya category of a manifold by projecting it onto a manifold of smaller dimension.

In this direction we would like to suggest that perverse Schobers are the right coefficient data for defining Fukaya categories. That is, to a perverse Schober S on a Kähler manifold X there should be naturally associated a triangulated category F(X, S), which for the constant Schober Db (VectC ) reduces to F(X). If we think about perverse sheaves in terms of some vector space data associated to Lagrangian cuts and then categorify these data to define Schobers, then it is natural to try to define F(X, S) in terms of these categorified data. For example, when X is a Riemann surface with marked points, the Fukaya category of X with coefficients in a constant (Z/2-graded) triangulated category A was defined in [16]. One can easily modify this definition when A is replaced by a local system of triangulated categories on X, i.e., by the next simplest instance of a perverse Schober. We leave the case of an arbitrary perverse Schober on a Riemann surface (§3E) for future work.

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Conventions.

For the considerations of this paper to make sense, we should work in some framework of “refined”, or “enhanced” triangulated categores having functorial cones. Let us describe one such framework, to be followed in the main body. (Another one would be that of stable ∞-categories [29].)

All our categories will be k-linear, where k is a fixed base field. We recall that Tabuada [36, 37] has introduced a Morita model structure on the category of dg-categories. Dg-categories fibrant with respect to this model structure are called perfect, see also [38]. Perfect dg-categories form a subclass of pre-triangulated categories in the sense of [11]. That is, a perfect dg-category A gives rise to a triangulated category H 0 (A) which is, in addition, idempotent complete. In the main body of this paper the word “triangulated category" will always mean “a triangulated category D together with an identification D H 0 (A) where A is a perfect dg-category", and “exact functor" will mean “an exact functor of triangulated categories obtained, by passing to H 0 , from a dg-functor between perfect dg-categories" and similarly for “natural transformation". With this understaning we will speak about “the” exact functor Cone{T : F ⇒ G} where T is a natural transformation of exact functors. For a finite CW-complex Z we denote Db (Z) the bounded derived category of all sheaves of k-vector spaces on Z. For a smooth complex b (X) the bounded derived category algebraic variety X we denote by Dcoh of coherent sheaves on X. The condition of perversity for constructible complexes on a complex manifold X is normalized in such a way that a constant sheaf in degree 0 is perverse. M.K.: Kavli Institute for Physics and Mathematics of the Universe (WPI), 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan, mikhail.kapranov@ipmu.jp

Vadim Schechtman Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France, schechtman@math.ups-toulouse.fr

References [1] R. Anno Spherical functors, arXiv:0711.4409.

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[2] R. Anno. Affine tangles and irreducible exotic sheaves, arXiv: 0802.1070. [3] R. Anno, T. Logvinenko. Orthogonally spherical objects and spherical fibrations, arXiv:1011.0707. [4] R. Anno, T. Logvinenko. Spherical DG-functors, arXiv:1309.5035. [5] M. Salah Bauendi, P. Ebenfeld, L. P. Rotschild. Real Submanifolds in Complex Space and Their Embeddings. Princeton Univ. Press, 1999. [6] A. Beilinson. How to glue perverse sheaves. In: K-theory, arithmetic and geometry (Moscow, 1984), Lecture Notes in Math. 1289, Springer-Verlag, 1987, 42 - 51. [7] A. Beilinson, I. Bernstein, P. Deligne. Faisceaux Pervers, Astérisque 100, 1982. [8] A. Beilinson, R. Bezrukavnikov, I. Mirkovic. Tilting exercises. Mosc. Math. J. 4 (2004), 547–557. [9] R. Bezrukavnikov, S. Riche. Affine braid group actions on derived categories of Springer resolutions. Ann. Sci. Éc. Norm. Supér. 45 (2012), 535-599. [10] A. I. Bondal, M. M. Kapranov. Representable functors, Serre functors and mutations. Math. USSR Izv. 35 (1990), 519-541. [11] A. I. Bondal, M. M. Kapranov. Enhanced triangulated categories. Math. USSR Sbornik, 70 (1991) 93-107. [12] P. Bressler, Y. Soibelman. Mirror symmetry and deformation quantization. arXiv hep-th/0202128. [13] C.Brav, H.Thomas, Braid groups and Kleinian singularities, Math. Ann. 351 (2011), 1005 - 1017. [14] Neil Chriss, Victor Ginzburg, Representation Theory and Complex Geometry. Birkhäuser, 1997. [15] J. Curry. Sheaves, Cosheaves and Applications. arXiv:1303.3255. [16] T. Dyckerhoff, M. Kapranov. Triangulated surfaces in triangulated categories. ArXiv: 1306.2545.

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[17] U. Frauenfelder, K. Zehmisch. Gromov compactness for holomorphic discs with totally real boundary conditions. arXiv:1403.6139. [18] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono. Lagrangian Intersection Floer Theory. Amer. Math. Soc. Publ. 2009. [19] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble, 35 (1985), 1-48. [20] S. Gelfand, R. D. MacPherson. Verma modules and Schubert cells: a dictionary. in: “Paul Dubreil and Marie-Paule Malliavin Algebra Seminar", 34th Year (Paris, 1981), pp. 1-50. Lecture Notes in Math. 924, Springer, Berlin-New York, 1982. [21] S. Gelfand, R. D. MacPherson, K. Vilonen. Perverse sheaves and quivers. Duke Math. J. 3 (1996), 621-643. [22] R. Harvey. The theory of hyperfunctions on totally real subsets of a complex manifold with application to extension problems. Amer. J. Math. 91 (1969), 853-873. [23] N. Honda, P. Schapira. A vanishing theorem for holonomic modules with positive characteristic varieties. Publ. RIMS Kyoto Univ. 26 (1990), 529-534. [24] M.Kapranov, V.Schechtman, Perverse sheaves over real hyperplane arrangements, arXiv:1403.5800, Ann. of Math. (2016), to appear. [25] M. Kashiwara, P. Schapira. Sheaves on manifolds. Springer, 1990. [26] M. Khovanov, R. Thomas. Braid cobordisms, triangulated categories, and flag varieties, arXiv:math/0609335. [27] A. Kuznetsov, V. Lunts. Categorical resolution of irrational singularities, arXiv:1212.6170. [28] G. Lebeau. Annulation de la cohomologie hyperfonction de certains modules holonomes. C.R. Acad. Sci. Paris, Sér. A, t.290 (1980), 313-316. [29] J. Lurie. Stable ∞-categories, arXiv math/0608228.

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Morfismos, Vol. 22, No. 2, 2018, pp. 121–173 Morfismos, Vol. 22, No. 2, 2018, pp. 121–173

Sheaf of categories and categorical Donaldson theory Sheaf of categories and categorical Donaldson Ludmil Katzarkovtheory Yijia Liu Ludmil Katzarkov

Yijia Liu

Abstract In this paper we take a new look at categorical linear systems applying the technique of sheaves Abstract of categories. We combine this technique withpaper the theory of acategorical metrics linear in order In this we take new look Kähler at categorical systems to build two parallels: applying the technique of sheaves of categories. We combine this 1) technique A parallel with with the Donaldson theory of Kähler-Einstein mettheory of categorical Kähler metrics in order rics. to build two parallels: 2) A parallel with Donaldson theory theory of polynomial invariants. met1) A parallel with Donaldson of Kähler-Einstein As rics. an outcome we introduce sheaves of categories which cannot be connected potentials obstructions to that are the moduli 2) Atoparallel withand Donaldson theory of polynomial invariants. spaces of stable objects. Connections sheaves categories withcannot As an outcome we introduce of sheaves of of categories which Homological Mirror to Symmetry non-complete andmoduli be connected potentialsforand obstructionsintersections to that are the the procedure arborealization are discussed as well. of categories with spaces ofofstable objects. Connections of sheaves Homological Mirror Symmetry for non-complete intersections and 2010 Mathematics Subject Classification: 14A22, 14J33, 53D37. the procedure of arborealization are discussed as well.

Keywords and phrases: Sheafs of categories, Donaldson theory. 2010 Mathematics Subject Classification: 14A22, 14J33, 53D37. Keywords and phrases: Sheafs of categories, Donaldson theory.

Introduction

The theory of linear systems is 2000 years old. Recently a new read of 1 Introduction this theory was suggested by the authors in [18]. Based on the recent The theory of by linear systems is 2000 Kontsevich, years old. Recently a new read of breakthrough made Haiden, Katzarkov, Pandit [15], who this theory was suggested by the authors in [18]. Based on the recent introduced the theory of categorical Kähler metrics, we develop further breakthrough made linear by Haiden, Katzarkov, Kontsevich, Pandit [15], who the theory of categorical systems. categorical Kähler metrics,linear we develop further In introduced this paper the we theory take a of new look at the categorical systems the theory of categorical linearofsystems. applying the technique of sheaves categories (see section 3). We In this paper we new look at the categorical linear systems combine this technique withtake the atheory of categorical Kähler metrics in the parallels: technique of sheaves of categories (see section 3). We order applying to build two combine this technique with the theory of categorical Kähler metrics in 1. A parallel withtwo Donaldson order to build parallels:theory of Kähler-Einstein metrics - [9]. 1. A parallel with Donaldson 121 theory of Kähler-Einstein metrics - [9]. 121

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2. A parallel with Donaldson theory of polynomial invariants - [8]. We briefly recall the theory of categorical Kähler metrics developed in [15]. We represent this work in terms of a categorical DonaldsonUhlenbeck-Yau correspondence outlined in the table below. We start with the classical GIT theory of the action of a group G with a unitary subgroup K on a manifold X. X/K

X/G Φ−1 (0)

Kempf-Ness Functional

The categorical interpretation of the above classical GIT setup, as followed in [15], can be presented as follows: C0

SC stable objects stable Lagrangians

The theory developed in [15] is based on the following correspondences: Table 1 C

X/G

X/K

equivariant line bundle

Φ : X → KV

M et(E)

G/K

Flow

pullback of grat Φ to G/K

Mass M

M < |Z|

Bogomolny condition

We briefly explain the above categorical setup in section 5. In his seminal work [7] Donaldson used the non-rationality of Dolgachev surfaces in order to find examples of homeomorphic but nondiffeomorphic surfaces. After developing the theory of sheaves of categories we propose a categorical interpretation of Donaldson’s results.

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We start with replacing the classical blow-up formulas in Donaldson theory with a blow-up formula of semi-orthogonal decompositions. Conjecture 1.1. Let C =< C1 , C2 > be a semi-orthogonal decomposition of a triangulated category C by admissible subcategories C1 , C2 . (Typical examples we have in mind are the bounded derived categories of coherent sheaves and Fukaya-Seidel categories - see e.g. [2].) Let Φ be a categorical version of Donaldson polynomial invariants (see section 7). Then (1)

Φ(C) = Φ(C1 )· Φ(C2 )· T (C),

where T (C) is a standard term. In section 7 we discuss wall crossing issues connected with the above conjecture. We consider some far-going applications of this conjecture in section 7. Using the categorical Kähler metrics we define canonical categorical Kähler-Einstein degenerations. We formulate: Conjecture 1.2. Let X0 ∪Xi be a canonical categorical Kähler-Einstein degeneration. Then we have the following equality defining the soul of polynomial invariants: (2)

(X0 ∪ Xi ) = Φ(X0 )· T (C). A major building block for applications is the following:

Conjecture 1.3. Let LG be a part of a four-dimensional LG model which contains all vanishing cycles corresponding to all cohomologies but h0,0 , h4,4 . Assume that these vanishing cycles produce b+ 2 ≥ 1 cohomologies of a minimal symplectic fourfold. Then LG = LG1 #LG2 , where each LGi is a LG model with b+ 2 ≥ 1 mirrors of minimal symplectic fourfolds. At the moment the conjectures formulated above are rather vague. In section 7 we outline some possible applications in the case of Fukaya categories. The considerations above suggest that we have a parallel: Classical

Categorical

Donaldson invariants

categorical invariants

non-diffeomorphic manifolds

non-rational varieties

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This parallel suggests that categorical Donaldson polynomial invariants are very deep invariants which measure the gluing of categories. In section 7 we spell out the connection between categorical basic classes and gaps in Orlov spectra as well as the connection between basic classes and dynamical spectra of the corresponding Fano manifolds. We propose that categorical Donaldson invariants define new birational invariants and we develop two techniques for studying the theory of categorical Donaldson invariants - categorical linear systems and the theory of sheaves of categories. There might be better ways of doing that. We discuss different applications of the idea of shaves of categories related to the proof of Homological Mirror Symmetry (HMS) for non-complete intersections. An important outcome of this paper is the following correspondence. Classical Donaldson Theory

Categorical Donaldson Theory

X = X1 #C X2

sheaf of categories is not connected with a potential

nontrivial moduli spaces

nontrivial moduli spaces of stable objects

wall crossing on metrics

wall crossing recorded in sheaves of categories

The paper is organized as follows. In section 2 we recall the theory of categorical linear systems. In section 3 we develop the theory of sheaves of categories. In section 4 we introduce the notion of categorical Kähler metrics. In section 5 we build the theory of categorical Okounkov bodies and in section 6 categorical Kähler-Einstein metrics. In section 7 we consider further applications.

Categorical linear systems

In this section we define two new notions - categorical linear systems and categorical base loci. We try to indicate the potential of these new notions for studying categories by connecting them with well-known categorical notions - gaps of spectra and phantoms. Let T be a saturated dg-category. Consider the endofunctors A, F of T . Definition 2.1. A noncommutative linear system is a collection of morphisms s ∈ Hom(A, F ). A pair of morphisms, s1 , s2 ∈ Hom(A, F ), is a noncommutative pencil.

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We may think of these morphisms as natural transformations si : A → F . We also define: Definition 2.2. The scheme-theoretic base locus of a noncommutative linear system C is the full subcategory of objects of T on which all s ∈ C vanish in the homotopy category. The triangulated base locus of C is the full subcategory of objects of T on which all s ∈ C act nilpotently in the homotopy category. Consider the case where X is an algebraic variety, T = Db (X) and F corresponds to tensoring with a line bundle. In this case, a noncommutative linear system abuts to the classical notion of a linear system by taking the homotopy classes of these morphisms. The scheme-theoretic base locus is precisely the full subcategory of complexes such that the cohomology is scheme-theoretically supported on the base locus. This is not a triangulated category. On the other hand, the triangulated base locus is the full subcategory of objects on which the cohomology is set-theoretically supported on the base locus. This is a triangulated category. Now, let L be any object of T . Consider RL,F = RHom(L, F n (L)). n=0

We consider every r in RHom(L, F n (L)), n > 0 as a morphism of graded bimodules r : RHom(L, F n (L)) → RHom(L, F n (L))[i]. We define Tors(RL,F ) to be the full subcategory consisting of all objects T in grmod over RL,F such that for every r in RL there exists N >> 0 such that rN (T ) = 0. Finally we define DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ). Definition 2.3. We define Tors(RL,F ) to be the L-base locus of the functor F .

The definition above is very complex and suggests that the categorical base locus measures the complexity of the functor F for a reasonable choice of the object L. In the case where F is a twist by a very ample line bundle on a smooth projective variety X and L is a line object (see e.g. [10]) in Db (X), we get that DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ) is just Db (X). For Artin-Zhang twists we get some noncommutative deformations of Db (X). But for more general functors some new phenomena appear in this categorical setting.

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The approach we suggest records the categorical base loci via marking divisors at infinity of the LG model, where we take the point of view that functors with high exts lead to bigger gaps in the Orlov spectra. In some cases the last phenomenon is recorded by the monodromy of the LG model. When working with mixed variations of stability structures we note that nontrivial exts in DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ) are connected with ghost sequences of length equal to the nontrivial exts. The categories DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ) and Tors(RL,F ) behave well under the following operations: 1. Birational maps. 2. Taking invariant or anti-invariant parts or combining F with any Schur functor. 3. We can modify Tors(RL,F ) to be defined as a full subcategory consisting of all objects T such that for every r in RL,F there exists N < ki such that rN (T ) = 0. So we have Tors(RL,F )k1 ⊂ Tors(RL,F )k2 for k1 < k2 . 4. Pencils, nets as well as fibrations of categories can be defined by choosing sections in DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ). Using the ghost sequences of the base categories (Db (P1 ), Db (P2 ) and so on) we obtain ghost sequences for DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ). 5. Assume that the functor F splits as a product of functors F = Fm · · · F1 . then we have RL,F ⊂ RL,Fm · · · RL,F1 . This formula will be implemented as the main ingredient of the K-calculus. Both formulas provide us with the opportunity to “glue” ghost sequences in order to calculate Orlov spectra. We give some simple examples.

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Table 2 P2p

•

(−KP2 − p)

I8 Cat. Base Locus

B = (Im F )⊥

1. Consider P2p - the image of P2 by the anti-canonical system with one base point p. Consider its mirror - an elliptic fibration with 4 fibers with usual double points singularities and an I8 fiber - see Table 2. The category generated by the image of the thimble vanishing in the fourth singular fiber in the generic open elliptic curve is the categorical base locus for the functor rotation around infinity. This is a simple consequence of Homological Mirror Symmetry - see e.g. [2]. There are two ways we can think of the creation of this base locus: 1) We localize FS category of the LG model for P2p by one thimble corresponding to the point p. (We return this singular fiber to infinity.) 2) We mark the point on the circle configuration of rational curves I9 . Both of these correspond to creating classical base loci of the linear system −KP2 − p. So by analogy with the classical situation we will think of this base locus as the marking of a point on the fiber at infinity. This marked point (this localized thimble) becomes a base point, categorical base locus, for the functor twist by −KP2 − p - the rotation around the fiber at infinity with the marked point fixed. We can think of this point as slightly moved from infinity but still close to infinity. The rotation functor keeps it fixed. 2. The example above can be interpreted as a projection functor. In general projection functors produce many examples of categorical liner systems many of which are new non-classical examples. Partial rotations in LG models also provide such examples.

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3. Noncommutative Lefschetz pencils: a) Week notion - two natural transformations s, t from the identity functor to the functor F . This does not give a functor from Db (P1 ) to the category T , because there is no requirement that st = ts. This notion produces a P1 -family of noncommutative divisors (the linear combinations of s and t) and the notion of base locus (objects on which s and t both vanish). b) The stronger notion requires a functor from Db (P1 ) to the category T , which amounts to the requirement that all the natural transformations considered must commute with each other. c) An intermediate notion which does not require st = ts, but only that st and ts agree up to some multiplicative factor - for 3 natural transformations r, s, t we just ask that the 9 natural transformations r2 , s2 , t2 , rs, st, tr, rt, ts, sr satisfy 3 linear relationships so that their span has rank 6 (as in a noncommutative P2 ). By analogy with the classical situation we will call the pencils from c) topological and the pencils from b) algebraic. We will work with algebraic noncommutative systems mainly. For a pictorial explanation of categorical base loci for Fukaya-Seidel categories - see Table 3. We can think of natural transformations of rotation functors and the identity functor as paths around the fiber at infinity. Intersections of these paths are the categorical base loci - the thimbles we have localized by. The geometry of this marked set plays an important role in our considerations.

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Table 3: Categorical Base Loci localized thimble

natural transformation •

I9 marking

•

∞

Base Loci

t = ⊗(−KP2 )

localization

marking

Functor

•

∞

Blowing up this base locus corresponds to creating a fiber at the LG model - see [2]. We move now to the definition of categorical multiplier ideal sheaf. Classically multiplier ideal sheaf is defined as follows. For a projective variety X and a linear system of the divisor D we define Jλi (D) = µ ∗ (OY (KY /X − λi µ∗ Σi si · Ei )), where si · Ei are divisors in the exceptional loci. We obtain the classical multiplier ideal sheaf by resolving singularities and taking the floor function, corresponding to taking parts of these divisors. As a result Jλi (D) measure singularities of the pair (X, D). We define the categorical multiplier ideal sheaf based on the approach via (3)

DGProj(RL,F ) = grmod over RL,F / Tors(RL,F )

developed above. We consider categorically a sequence of functors λi F acting on modified categories C i defined as (4) J(C i , λi F ) = DGProj(RLi ,λi F ) = grmod over RLi ,λi F / Tors(RLi ,λi F ).

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If the functor F = F λk · · · F λ1 , then λi F = F λi−1 · · · F λ1 .

Definition 2.4. We define the sequence of categories J(C i , λi F ) to be a categorical multiplier ideal sheaf. In the case of X being a smooth projective variety, L a line object (e.g. OX ) and F a twist by an ample line bundle RL,λi F we have an analogue to the classical multiplier ideal sheaf situation. Indeed if F is a twist by a divisor D = D1 + · · · + Dk = Σi si · Ei we get a functor F = Fk · · · F1 . This observation suggests a generalization - the definition of categorical multiplier ideal sheaf. Assume that Fi commute. We get RL,F ⊂ RL,Fk ···F1 with the corresponding sequence of categories. From this prospective: (5)

F = Fk , λk−1 F = Fk · Fk−1 , . . . , λ1 F = Fk · · · F1 .

So the categorical multiplier ideal sheaf is defined by (6) J(C i , λi F ) = DGProj(RLi ,λi F ) = grmod over RLi ,λi F / Tors(RLi ,λi F ) as a sequence of localizations which measure the complexity of the functor F . Classically for the mixed Hodge structure (MHS) associated with the function f defining D we have a spectrum of the MHS (see [5]). The monodromy ei.λi of the MHS of f is connected with the classical multiplier ideal sheaf J(X, λi F ). By analogy with [5] we conjecture that there exists a matrix factorization category M F so that the spectrum of the mixed noncommutative Hodge structure associated with it produces the jumping numbers of the categorical multiplier ideal sheaf. We give as an example the following theorem - see [18]. Theorem 2.5. The multiplier ideal sheaf for the category An and the localization functor - restricting to An−1 determine the Orlov spectrum of An . In this case the categorical multiplier ideal sheaf is a sequence of localizations. We plan to compute more examples. One important case is the so-called LG functor. Definition 2.6. We will call F a LG functor of the Fukaya-Seidel category if F is a functor of a rotation around the fiber at infinity and some other fibers of the LG model associated with the mirror of a smooth projective variety. This means that these fibers are unchanged under the rotations see Figure 1.

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Figure 1: Canonical LG functors

•

∞

Examples of LG functors are the A side realizations of the Serre functor. In most of the examples in this paper we will consider the mirrors of smooth Fano manifolds. Most of the LG models associated with the mirror of a smooth projective Fano manifold have a singular fiber at infinity (this is certainly true for non-rational Fano manifolds). Definition 2.7. We will call F a canonical LG functor of the FukayaSeidel category of the mirror of a smooth Fano manifold if F is a functor of a rotation around the fiber at infinity and all other nonzero singular fibers. In other words all fibers but the zero fiber is left fixed or this is a rotation around the fiber at zero. Building the theory of categorical multiplier ideal sheaves for LG functors and especially for canonical LG functors is the main part of this project. The categorical multiplier ideal sheaf (7)

J(Ck , λk F ) ⊂ · · · ⊂ J(C1 , λ1 F )

in the case of LG functors is a sequence of localizations J(Ci , λi F ). By analogy with the An case we start with the localization by the biggest thimble and we obtain J(Ck , λk F ) - this corresponds to marking the whole singular fiber over zero. We start unmarking this singular fiber so that the rank HP (T ) of the category T we localize by goes down by one. J(Ck , λk F ) J(Ck−1 , λk−1 F ) F 0

J(Ck−2 , λk−2 F )

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Similarly we can define the categorical multiplier ideal sheaf J(Ck , λk F ) ⊂ · · · ⊂ J(C1 , λ1 F )

(8)

for the derived category of singularities. In this case localizations are nothing else but blowing-down components of the central fiber, A deeper analysis of DGProj(RL,F ) = grmod over RL,F / Tors(RL,F ) and Tors(RL,F ) supports these expectations.

Real blow-ups, sheaves of categories and linear systems

In this section we develop a connection between sheaves of categories and LG models. We first define a sheaf of categories. Our treatment is parallel to [17] but somewhat different. Definition 3.1. Let Graph be the incidence category of a graph Γ. Let F be a topological local system of categories. We call (9)

Sheaf (F) := lim F unc(Cgraph → F), −→

g ∈ Graph

a simple sheaf of categories, where g is the incidence graph of the embedded in Cgraph . g

•

Definition 3.2. Let Sheaf (F1 ) and Sheaf (F2 ) be two simple sheaves of categories and BlowS 1 be a real blow-up of C. BlowS 1 R

Sheaf (F1 ) Sheaf (F2 ) Φ : Sheaf (F2 ) Sheaf (F2 ) functor of vanishing cycles We call Sheaf (F1 ) R Sheaf (F2 ) a perverse sheaf of categories over C glued by the functor R. Example 3.3.

1. 1-dim LG model

Sheaf of categories and categorical Donaldson theory

1-dim LG model D F (E) ⊗ A3 •

(P2 )

−→ F S(C∗ 2 , x + y +

F (E) •F (E)/ C

•

• • •

P S1 F (E) • F (E)

•

real blow-up

•

1 xy )

•

leads to the gluing procedure for a spherical functor S 1 F (E) spherical functor • Φ F (E) φ • O DS 1 gluing DO Φ F (E)/Γ = F (E)/ Ker φ

2. 2-dim LG model

133

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2-dim LG model S1 L1

•

L1 O

O L2 gluing

D 2 × S1

D01

D11

S1 × S1

D00

D10

gluing S1 × D

Example 3.4. We give one more example of applications of sheaves of categories the example of stability Hodge structures. Quiver •

dual curve

C2 /(xy − 1)

•

C The general picture is: JC Γ ·

Γ •

•

Quiver

• dual curve MΓ Hn,d hyperplane arrangements

The above MΓ can be used as a LG mirror of the category of representations of the quiver Γ.

Conjecture 3.5. Ck /∪Hd,n is the moduli space of stability conditions.

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Definition 3.6. We call H : JC Γ

Ck / ∪ Hd,n

Cn a stability Hodge structure. Theorem 3.7 (Torelli). H recovers category Γ. Proof. It follows from the definition of sheaves of categories in multidimensional LG models. Corollary 3.8. H : JC Γ −→ Ck / ∪ Hd,n recovers the Orlov spectrum of Γ. For An and An / < Γ1 , . . . , Γp >, we checked this conjecture in [18], where a detailed account of Orlov spectra and its gaps are given.

Hodge diamond

•

LG1 vanishing cycles LG1 • •

• • •

• •

•

LG2

LG2 •

• • •

∞

The theory of sheaves of categories suggests the following: Definition 3.9. Let LG1 be a part of a LG model which contains all vanishing cycles corresponding to all cohomologies but h0,0 , hn,n . We call LG1 #LG2 the min topological sum of LG models. One of the most celebrated example of a LG model (non)stretching is the example of Dolgachev surface worked out by Donaldson in [7]. We interpret Donaldson’s result from the point of view of sheaf of categories. We start with the change of the sheaf of categories structure on rational elliptic surfaces to Dolgachev surface - see below.

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Table 4: Log transform •

•

2 dim LG •

•

lo g(

•

•∞

3) • •

• •

4 dim LG ∞

We give some more examples of the splitting of LG models. We start with the example of Hirzebruch surface F 1 .

LG1

•

f : P1 −→ M st (CY ) LG2 LG1 #LG2 •

Discr Example 3.10 (LG(F 1 )).

...... neck special metric stab condition •••

LG1 #LG2 M (CY )

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Example 3.11 (LG of F 1 = C P2 #C P2 ). •

Fuk(E)/Γ •

• •

A4 ⊗ Fuk(E)

•

w→∞

• •

∞

•

w→0

•

stretching P1

LG1

A3 ⊗ Fuk(E) • Fuk(E) • •∞ • •

•

• •

w→∞

LG2

LG=LG1 #LG2 An important observation is the following conjecture. Conjecture 3.12. The sheaf of categories associate with Dolgachev surface is not connected with a potential. This is an analogue of Donaldson’s statement that Dolgachev surface is not a connected sum of two 4-dimensional manifolds. Similarly we can introduce the stretching in a 2-dimensional LG model too. The procedure of stretching the neck is the resolving of the singularities of the curves in the base of the 2-dimensional LG model. Creating obstructions to stretching as in the classical Seiberg-Witten theory can be done with surgeries - changing the Alexander polynomials - see [14]. We briefly describe the procedure below. Example 3.13 (2-dim LG model). LG w → ∞

LG2 w→0

LG1 C2

w→∞

LG1

LG2

Observe that being a “stretched neck” is more than a semi-orthogonal decomposition. So to measure this we take Donaldson’s point of view use moduli spaces of objects. We suggest the parallel in conjecture 1.3.

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A proof of this statement will lead to building an analogy between Donaldson theory and categorical Kähler metric. We summarize the analogy in the table below: Table 5 Categorical Donaldson theory Donaldson Theory µ → H2 (X) H (B/ASD connections) − B - moduli space of ASD connections 2

M - compact moduli space

Donaldson Γ-invariant Γ() = 2µ(M ) + c1 (KX ) Categorical theory Classical

Categorical

B H (B) −→ H2 (X) ∪ Chamber structure

M et = {E, h} M et −→ Stab(C) reducible metrizable object

Γ = µ(M ) + c1

limiting HN filtrations ∪ Chamber structure Γ = µ(M ) + c1

We also build the parallel with Bridgeland’s theory of stability conditions.

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Table 6: Example Dolg 2, 3 2 p1 ,...,p9 Y =P Γ() = c1 (KY )

... 2

Z - Dolg 2, 3

Γ() = c1 (KZ ) + 7c1 (KZ )

• |

H = H10

Stab

H0 + H4 chambers 2

Stab

+ 0

+ 4

Classical Donaldson theory Chamber structures and reduced connections

reduced metrizable object= phantom Γ-invariant non-rationality in general

Γ-invariant non-rationality for surfaces

The approach with sheaves of categories can be useful in many directions. We give two applications connected with HMS. We start with the following categorification of classical Enrique-Petri theorem, see [11]. Let C be a trigonal canonical curve, g(C) = g. In Pg−1 , we have (10)

C⊂S=

c⊂Qi

here Qi are quadrics in Pg−1 and S is the ruled surface. We describe a procedure of building HMS for genus g curve which is mod a complete intersection.

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Step 1. We degenerate S to the union of P1 × P1 .

S 1 P1 × P1

S n S n−1

Step 2. Now we apply [1] to each S 1 . We get sheaves of categories.

FS(LG(S1 ))

FS(LG(S2 ))

······ S1

Step 3. Then we regenerate gluing sheaves of categories and using [1].

•

We get the HMS for C. Similarly we can do it for any canonical curve using Enrique-Petri theorem. Theorem 3.14. The procedure above proves HMS for canonical curves. The procedure above can be seen as a part of a much more general procedure of arborealization introduced by Nadler [22]. Let X be a 3-dim Fano and LG1 (W → C) is 1-potential LandauGinzburg model and LG2 (W → C2 ) is Landau-Ginzburg with two potentials. Theorem 3.15. The transform from sheaf of categories LG1 (W → C) to LG2 (W → C2 ) is a sheaf of categories version of arborealization.

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For more see [22] and [14].

Arborealization

Categorical Kähler metrics

In this section we briefly describe the categorical Kähler metric. The notion was developed in [15]. We recall the definitions and explain Table 1 from the introduction. The idea of Kähler metric of a category imitates the classical definition up to a point. Let C be a triangulated category. We define a new category C 0 . ⊗O K

Object of C 0 := E, h :−−−K−→ E, where h is a metric on E. (11)

M et(E, h) = E ∈ Ob C 0 + iso h : E ⊗OK K ∼ E

We give a definition of categorical Kähler metric - it is a bit of cloud which allows us to define moduli spaces of objects. Definition 4.1. Let C be a triangulated category and let Z : K(C) → C be a central charge. We define the moduli space M et(E, h) as above. We say that C is a categorical Kähler metric if C is enhanced with the following data: D(1) Function M ass : M et(E, h) → R. D(2) Flow F : M et(E, h) → R. D(3) Two other functions Amp− ≤ Amp+ : M et(E, h) → R. D(4) Potential function S : Ob(C 0 ) → C. (additional care is needed to make data from different t-structures compatible and make M et(E, h) a nice space) satisfying the following axioms:

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A(1) The triangulated structure on C is recorded by the following actions: Z-shift, R 0 -flow, R-rescaling. They form a group Z (R 0 × R). A(2) These actions are compatible with the functions Mass, Amp− , Amp+ , S. A(3) Additivity of all these functions on M et(E, h), namely: Z((E, h), (F, h)) = Z + Z; S C (⊕) = S C + S C ; Amp− (⊕) = min(Amp− ,Amp− ); Amp+ (⊕) = min(Amp+ ,Amp+ ). A(4) Qualitative properties - fixed points of the flow. Mass(E, h) Z(E) Mass= |Z(E)| ↔ ∃! θ ∈ R, F - θR = 0 on (E, h). Then E is stable.

∀(E, h), lim eF t (E, h), ∃ in the flag compactification t→∞

(E1 , h1 ), . . . , (En , hn ). ∀i, ∃! θi , θ1 > · · · > θn , (F − θi R)|(Ei ,hi ) = 0. We call these fixed points Harder–Narasimhan filtrations. A(5) There exists a compactification of the moduli space of Z-stable objects which consists of HN filtrations. These axioms were used in [15] to prove the existence of compact moduli spaces of stable objects. We will use these moduli spaces to build categorical Donaldson invariants.

Categorical base loci and categorical Okounkov bodies

In this section we introduce the notions of categorical linear systems and categorical base loci. We connect these notions with gaps of spectra of categories. From what we have said it becomes clear that Okounkov bodies play an important role in studying the complexity of functors. In order to

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make it suitable to our noncommutative birational approach we devise a categorical analogue of the notion of Okounkov body ∆(D). Indeed, the notions of divisors and their sections and multiplicities can be immediately translated into categorical language. The flag of submanifolds becomes a flag of subcategories. The sections become natural transformations between spherical and restriction functors, and valuations νi correspond to how far along one can lift these natural transformations see the Figure 2 and Figure 3 below. In these figures S is the restriction functor and t is a spherical functor of a twist by a divisor. ···

{−3R1 }

{−2R1 }

{−R1 }

Id t {−D}

Figure 2: Step 1 (ν1 = k1 = 2) Definition 5.1. Following the figure above we define ν1 as the maximal number of liftings of the natural transformation t. ···

{−3R2 }

{−2R2 }

{−R2 }

IdR1 t {−k1 R1 − D}

Figure 3: Step 2 (ν2 = k2 = 3) Similarly we define ν2 as the maximal number of liftings in the figure above. In the same way we define νi using the flag of subvarieties R1 , ..., Rd or of subcategories. Remark 5.2. Definition 5.1 is a categorification of the usual definition of Okounkov body. Classically ki is the multiplicity with which D passes through Ri . We give several examples of flags of categories mainly coming from derived categories of flags of subvarieties. The cube of categories below is given by two quadrics Q1 and Q2 in P3 and their intersection - an

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elliptic curve E. As shown in Figure 4, derived categories of Qi and E define a flag of categories R1 , R2 . Db (X)

Db (Q2 )

Db (Q1 )

Db (Q1 ∩ Q2 ) Db (Q3 )

Db (Q1 ∩ Q3 )

Db (Q2 ∩ Q3 )) Db (Q1 ∩ Q2 ∩ Q3 )

Figure 4: B side The Homological Mirror Symmetry defines equivalent Fukaya-Seidel (FS) categories (see e.g. [2], [20]) with a mirror cube given below in Figure 5 as well as the mirror of the flag of derived categories. FS(Z, w) FS(Z1 , w1 )

FS(Z2 , w2 ) FS(Z12 , w12 )

FS(Z3 , w3 ) FS(Z13 , w13 )

FS(Z23 , w23 ) Fuk(E)

Figure 5: A side Definition 5.3. Consider a flag of categories R1 , ..., Rd . For m ∈ Z>0 , denote t◦m by m· t. Now we can define the Im(m· t) as (ν1 , ..., νd ), where every νi depends on m· t. As a result we have: Definition 5.4. We define δ(t) as the closed convex hull of lim

m→∞

1 Im(m· t). m

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δ(t) is a categorical notion. We start with two functors. δ(t) measures how these two functors interact asymptotically with respect to a flag of subcategories. We will give more examples later. The categorical Okounkov body will play an important role in classifying the base loci of a category. Remark 5.5. Observe that we get some modifications of Okounkov body if we additionally twist divisor D by a multiplier ideal sheaf - test configuration. In fact we get a sequence of Okounkov bodies associated with the filtration on the sheaf of ideals. For more see [18].

Categorical multiplier ideal sheaves and categorical Kähler-Einstein metrics

In this section we present a scheme for building theory of categorical Kähler-Einstein metrics. We start with examples of categorical multiplier ideal sheaves. Categorical Multiplier Ideal Sheaf for An : One of the main observations of [18] is that the spectra of An can be interpreted from the point of view of categorical multiplier ideal sheaves for the category An . We have a sheaf of generators (a sheaf of localized categories) for which the jump numbers determine how many sides do we take from the whole polygon in order to form the forbidden part. We record our observation in the following theorem - see also Table 8. Theorem 6.1. The multiplier ideal sheaf for the category An and the 1 localization functor - restricting an n-gon has jump numbers n−1 n , . . . , n. The multiplier ideal sheaf J(λ1 , . . . , λk ) determines the Orlov spectrum of An . The proof of this theorem is a direct consequence of the definition of J(λ1 , . . . , λk ) - see Table 8. In this case the categorical multiplier ideal sheaf is a sequence of localizations J(Ck , λk F ) ⊂ · · · ⊂ J(C1 , λ1 F ). Marking a polygon corresponds to localizing by a subcategory. The localization by the biggest polygon produces the first non-trivial category J(Ck , λk F ) and by the smallest J(C1 , λ1 F ). In the table below we represent the multiplier ideal sheaf in this case as rotation by angles of λj of the localization functor F.

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Table 7 The Mult. Ideal Sheaf

Functor Localization by An D

Â·Â·Â·

(â&#x2C6;&#x2019;2p)

(â&#x2C6;&#x2019;p)

â&#x20AC;¢

J( nâ&#x2C6;&#x2019;1 n , D) â&#x20AC;¢

Î»j s

â&#x20AC;¢

j =nâ&#x2C6;&#x2019;1

â&#x20AC;¢

j =nâ&#x2C6;&#x2019;2

â&#x20AC;¢

D â&#x2C6;&#x2019;Î»j F

â&#x20AC;¢

â&#x20AC;¢ â&#x20AC;¢

â&#x20AC;¢

J( nâ&#x2C6;&#x2019;2 n , D) â&#x20AC;¢

â&#x20AC;¢

As the proof of Theorem 6.1 shows obtaining Orlov spectra is moving up on marking polygons - see the table below. This is the baby Kcalculus. Table 8 Spectra { nâ&#x2C6;&#x2019;1 2 } â&#x2C6;ª {n â&#x2C6;&#x2019; 1}

Sheaves and Jump numbers Î»n =

nâ&#x2C6;&#x2019;1 n

nâ&#x2C6;&#x2019;1 {( kâ&#x2C6;&#x2019;1 2 ), ..., 2 } â&#x2C6;ª{k â&#x2C6;&#x2019; 2, ..., n â&#x2C6;&#x2019; 1}

Î»k =

k n

{0, 1, . . . , n â&#x2C6;&#x2019; 1}

Î»1 =

1 n

As we have seen the Okounkov body and the multiplier ideal sheaf can be made totally categorical. (Showing that categorical valuations we have defined satisfy the usual equalities requires additional work see [14].) Classically these two notions have been used to define: 1. Futaki invariants - integral of functions (defining testing configurations) over Okounkov bodies for a line bundle L. If these integrals over all these testing configurations are all positive we conclude that the smooth projective variety X has a KaÌ&#x2C6;hlerâ&#x20AC;&#x201C;Einstein met-

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ric. 2. The log canonical threshold - the smallest jump number. If a Fano manifold X of dimension n has a log canonical threshold bigger than n/(n + 1) then we conclude that smooth projective variety X has a Kähler–Einstein metric (see [6], [23]). The categorical interpretation of the Futaki invariant is an integral over the categorical Okounkov body defined by a functor F (the categorical version of a twist by L) and a testing configuration - an additional twist by an ideal choosing faces of the categorical Okounkov body. We will call these special functors Landau-Ginzburg testing functors. We introduce the definition: Definition 6.2. We will call a category a Kähler–Einstein category if all categorical Futaki invariants for all Landau-Ginzburg testing functors are positive. Recall that: Definition 6.3 (Testing LG functors). We call a family of LG functors a testing configuration fk LGt

LGt C

LG0 f1 0

compatible with C∗ -action iff 1. ∀t = 0, LGt are isomorphic; 2. LG0 consists of several LG models. Example 6.4 (LG(P2 )).

f = f1 · · · f k

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f • •

•

• •

•

0 •

I(f )

•

function defining test configuration f = f1 · f 2

Categorical Futaki invariant = Mod(Object)= Mt LGt

∆(L) I(f )s

M1 M2 M3

LG0 |

t 0 φ(Mt ) → φ(M1 )φ(M2 )φ(M3 )

We collect the Kähler–Einstein correspondences in Table 9 below.

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Table 9 Classical Kähler-Einstein

Categorical Kähler-Einstein

X Fano, dim(X)= n