Microlocal sheaves and quiver varieties

Roman Bezrukavnikov; Mikhail Kapranov

doi:10.5802/afst.1502

Roman Bezrukavnikov ¹; Mikhail Kapranov ²

¹ Department of Mathematics, MIT, Cambridge MA 02139 USA
² Kavli IPMU, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583 Japan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 473-516.

Abstract
Abstract

We relate Nakajima Quiver Varieties (or, rather, their multiplicative version) with moduli spaces of perverse sheaves. More precisely, we consider a generalization of the concept of perverse sheaves: microlocal sheaves on a nodal curve $X$ . They are defined as perverse sheaves on normalization of $X$ with a Fourier transform condition near each node and form an abelian category $M (X)$ . One has a similar triangulated category $D M (X)$ of microlocal complexes. For a compact $X$ we show that $D M (X)$ is Calabi-Yau of dimension 2. In the case when all components of $X$ are rational, $M (X)$ is equivalent to the category of representations of the multiplicative pre-projective algebra associated to the intersection graph of $X$ . Quiver varieties in the proper sense are obtained as moduli spaces of microlocal sheaves with a framing of vanishing cycles at singular points. The case when components of $X$ have higher genus, leads to interesting generalizations of preprojective algebras and quiver varieties. We analyze them from the point of view of pseudo-Hamiltonian reduction and group-valued moment maps.

Nous relions les variétés de carquois de Nakajima aux espaces de modules des faisceaux pervers. Notamment, nous considérons une généralisation des faisceaux pervers : les faisceaux microlocaux sur une courbe nodale $X$ . Ils sont definis comme les faisceaux pervers sur le normalisé de $X$ satisfaisant une condition sur le transformée de Fourier. Ils forment une catégorie abélienne $M (X)$ . On a aussi une catégorie triangulée $D M (X)$ contenant $M (X)$ . Pour $X$ compacte nous prouvons que $D M (X)$ est une catégorie de Calabi-Yau de dimension 2. Dans le cas où toutes les composantes irréductibles de $X$ sont rationelles, $M (X)$ est équivalente à la catégorie des représentations de l’algèbre pré-projective multiplicative associée au graphe d’intersection de $X$ . Les variétés de carquois proprement dites sont obtenues comme espaces de modules des faisceaux microlocaux munis d’une paramétrisation des cycles évanescents aux points singuliers. Dans le cas où les composantes de $X$ sont de genre superieur, on obtient d’intéressantes généralisations des algèbres pré- projectives et des variétés de carquois. Nous les analysons du point de vue de la réduction pseudo-Hamiltonienne et des applications moment à valeurs dans un groupe.

Published online: 2016-07-11

MR Zbl

DOI: 10.5802/afst.1502

Author's affiliations:

Roman Bezrukavnikov ¹; Mikhail Kapranov ²

¹ Department of Mathematics, MIT, Cambridge MA 02139 USA
² Kavli IPMU, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583 Japan

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     author = {Roman Bezrukavnikov and Mikhail Kapranov},
     title = {Microlocal sheaves and quiver varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {473--516},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {2-3},
     year = {2016},
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Roman Bezrukavnikov; Mikhail Kapranov. Microlocal sheaves and quiver varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 2-3, pp. 473-516. doi : 10.5802/afst.1502. https://afst.centre-mersenne.org/articles/10.5802/afst.1502/

References
Cited by

[1] Alekseev (A.), Bursztyn (H.), Meinrenken (E.).— Pure spinors on Lie groups. Astérisque 327 p. 131-199 (1999).

[2] Alekseev (A.), Kosmann-Schwarzbach (Y.), Meinrenken (E.).— Quasi-Poisson manifolds. Canad. J. Math. 54 p. 3-29 (2002). | DOI | MR

[3] Alekseev (A.), Malkin (A.), Meinrenken (E.).— Lie group valued moment maps. J. Diff. Geom. 48 p. 445-495 (1998). | DOI | MR | Zbl

[4] Beilinson (A.).— How to glue perverse sheaves. In: $K$ -theory, arithmetic and geometry (Moscow, 1984), Lecture Notes in Math. 1289, Springer-Verlag, p. 42 - 51 (1987). | DOI | Zbl

[5] Beilinson (A.), Bernstein (J.).— A proof of Jantzen’s conjectures. In: I. M. Gelfand Seminar, 1-50, Adv. Soviet Math.116, Part 1, Amer. Math. Soc. (1993). | DOI | Zbl

[6] Beilinson (A.), Bernstein (J.), Deligne (P.).— Faisceaux Pervers. Astérisque 100. | Zbl

[7] Boalch (P. P.).— Global Weyl groups and a new theory of multiplicative quiver varieties. arXiv:1307.1033. | DOI | MR | Zbl

[8] Boalch (P. P.).— Geometry and braiding of Stokes data: fission and wild character varieties. Ann. Math. 179 p. 301-365 (2014). | DOI | MR | Zbl

[9] Boalch (P. P.).— Wild Character Varieties, points on the Riemann sphere and Calabi’s examples. arXiv:1501:00930. | DOI | MR | Zbl

[10] Bondal (A. I.), Kapranov (M. M.).— Enhanced triangulated categories. Mat. USSR Sb. 181 p. 669-683 (1990). | DOI | Zbl

[11] Boyarchenko (M.).— Quantization of minimal resolutions of Kleinian singularities. Adv. Math. 211 p. 244-265 (2007). | DOI | MR | Zbl

[12] Calaque (D.).— Lagrangian structures on mapping stacks and semi-classical TFTs. ArXiv: 1306.3235. | DOI

[13] Crawley-Boevey (W.).— Monodromy for systems of vector bundles and multiplicative preprojective algebras. Bull. London Math. Soc. 45, p. 309-317 (2013). | DOI | MR | Zbl

[14] Crawley-Boevey (W.), Shaw (P.).— Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem. Adv. Math. 201 p. 180-208 (2006). | DOI | MR | Zbl

[15] Ciocan-Fontanine (I.), Kapranov (M.).— Derived Quot schemes. Ann. École Norm. Sup. 34, p. 403-440 (2001). | DOI | MR | Zbl

[16] Etingof (P.), Ginzburg (V.).— Noncommutative del Pezzo surfaces and Calabi-Yau algebras. J. Eur. Math. Soc. 12 p. 1371-1416 (2010). | DOI | MR | Zbl

[17] Faonte (G.).— Simplicial nerve of an $A_{\infty}$ -category. Preprint arXiv:1312.2127. | MR | Zbl

[18] Fock (V.), Goncharov (A.).— Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. IHES, 103, p. 1-211 (2006). | DOI | Zbl

[19] Galligo (A.), Granger (M.), Maisonobe (Ph.).— D-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier (Grenoble) 35, p. 1-48 (1985). | DOI | MR | Zbl

[20] Ginzburg (V.).— Calabi-Yau algebras. Preprint arXiv math/0612139.

[21] Goldman (W. M.).— The symplectic nature of fundamental groups of surfaces. Adv. in Math. 54, p. 200-225 (1984). | DOI | MR | Zbl

[22] Kashiwara (M.).— Vanishing cycle sheaves and holonomic systems of differential equations, in: “Algebraic geometry (Tokyo/Kyoto, 1982)" p. 134-142, Lecture Notes in Math. 1016, Springer, Berlin (1983). | DOI

[23] Kashiwara (M.), Schapira (P.).— Sheaves on Manifolds. Springer (1990). | DOI

[24] Kashiwara (M.), Schapira (P.).— Deformation Quantization Modules. Astérisque 345.

[25] Kontsevich (M.).— Soibelman (Y.).— Notes on $A_{\infty}$ -algebras, $A_{\infty}$ -categories and non-commutative geometry. Preprint arXiv math/0606241. | DOI

[26] Lurie (J.).— Stable $\infty$ -categories. Preprint arXiv math/0608228.

[27] Lurie (J.).— Derived algebraic geometry II-V. Preprints arXiv 0702299, 0703204, 0709.3091, 0905.0459.

[28] Lurie (J.).— Higher Algebra, book in preparation.

[29] Malgrange (B.).— Equations Différentielles à Coefficients Polynomiaux. Progress in Mathematics 96, Birkhäuser Boston (1991). | Zbl

[30] Maulik (D.), Okounkov (A.).— Quantum groups and quantum cohomology. Preprint arXiv:1211.1287. | DOI | Zbl

[31] Milnor (J.).— Introduction to Algebraic K-theory. Princeton Univ. Press, (1971).

[32] Mukai (S.).— Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77 p. 101-116 (1984). | DOI | MR | Zbl

[33] Nakajima (H.).— Quiver varieties and Kac-Moody Lie algebras. Duke Math. J. 91 p. 515-560 (1998). | DOI | MR | Zbl

[34] Pantev (T.), Toën (B.), Vaquié (M.), Vezzosi (G.).— Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117 p. 271-328 (2013). | DOI | MR | Zbl

[35] Safronov (P.).— Quasi–Hamiltonian reduction via classical Chern-Simons theory. Arxiv: 1311.6249. | DOI | MR

[36] Sibilla (N.), Treumann (D.), Zaslow (E.).— Ribbon graphs and mirror symmetry. Selecta Math. 20, p. 979-1002 (2014). | DOI | MR | Zbl

[37] Tabuada (G.).— Théorie homotopique des dg-categories, preprint ArXiv 0710.4303. | DOI | MR

[38] Toën (B.).— The homotopy theory for dg-categories and derived Morita theory. Invernt. Math. 167 p. 615-667 (2007). | DOI | MR | Zbl

[39] Toën (B.).— Derived algebraic geometry. Preprint arXiv:1401.1044.

[40] Toën (B.), Vaquié (M.).— Moduli of objects in dg-categories.Ann. Sci. École Norm. Sup. (4) 40, p. 387-444 (2007). | DOI | MR | Zbl

[41] Toën (B.), Vezzosi (G.).— Homotppical algebraic geometry II: Geometric stacks and applications. Mem. AMS, 902, x+224 pp (2008). | DOI | MR

[42] van den Bergh (M.).— A relations between Hochschild homology and cohomology for Gorenstein rings. Proc. AMS 126 (1998) 1345-1348. Erratum, ibid. 130 p. 2809-2810 (2002). | DOI

[43] van den Bergh (M.).— Double Poisson algebras. Trans. Amer. Math. Soc. 360 p. 5711-5769 (2008). | DOI | MR | Zbl

[44] van den Bergh (M.).— Non-commutative quasi-Hamiltonian spaces. in: “Poisson geometry in mathematics and physics", Contemporary Math. 450 p. 273-299, Amer. Math. Soc. (2008). | DOI

[45] Yamakawa (D.).— Geometry of multiplicative preprojective algebras. Preprint arXiv:0710.2649. | DOI | MR | Zbl

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