2002/12/19 by Victor Ginzburg, Ginzburg, Victor, D. Kaledin +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.math/0212279
openalex publication_date 2002/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 G⊂ Sp(V) 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 rational cohomology H^*(X) of any smooth symplectic resolution X → V/G (multiplicative McKay correspondence). We prove further that if G is an irreducible Weyl group in GL(h) and V=h+ h^* then no smooth symplectic resolution of V/G exists unless G is of types A,B, or C.