Microlocal sheaves and quiver varieties
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 473-516.

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 DM(X) contenant M(X). Pour X compacte nous prouvons que DM(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.

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 DM(X) of microlocal complexes. For a compact X we show that DM(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.

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     author = {Bezrukavnikov, Roman and Kapranov, Mikhail},
     title = {Microlocal sheaves and quiver varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Bezrukavnikov, Roman; Kapranov, Mikhail. Microlocal sheaves and quiver varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 2-3, pp. 473-516. doi : 10.5802/afst.1502. http://archive.numdam.org/articles/10.5802/afst.1502/

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