Poisson deformations of symplectic quotient singularities

Abstract

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.In particular, let V be a finite-dimensional complex symplectic vector space and GCSpðV Þ a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V =G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology H ðX ; CÞ of any smooth symplectic resolution X 7V =G (multiplicative McKay correspondence). We prove further that if GCGLðhÞ is an irreducible Weyl group and V ¼ h"h à ; then no smooth symplectic resolution of V =G exists unless G is of types A; B; C: r

Théorie des opérades de Koszul et homologie des algèbres de Poisson

“…Braconnier [4], Huebschmann [24], Krasilshschik [28], Lichnerowicz [29]). Mentionnons que Ginzburg et Kaledin utilisent la cohomologie de Poisson dans leur travail sur les résolutions de singularités symplectiques [21].…”

“…On renvoie le lecteur à l'article de Ginzburg-Kaledin [21] pour cet aspect spécifique de la théorie. Nos calculs d'homologie, annoncés dans la note [14], seront rédigés dans un article ultérieur.…”

Hochschild cohomology and graded Hecke algebras

“…Fix a finite group G and a representation V = C n of G. The Hochschild cohomology of S(V )#G is given in [12,14] when G acts faithfully on V . In this section, we reformulate this result to aid our explicit computations and our determination of graded Hecke algebras.…”

“…The graded vector space structure of Hochschild cohomology HH q (S(V )#G) can be described in terms of invariants of centralizer subgroups of G (see [12,14]). We recall this structure in Section 3 and reformulate it in terms of semi-invariants.…”

On the (non)existence of symplectic resolutions of linear quotients

“…When the parameter c is zero, Z c (G) is the coordinate ring of V /G. It is known, by [11], that Z c (G) defines a flat Poisson deformation of V /G. …”