Poisson deformations of symplectic quotient singularities

By V. Ginzburg, D. Kaledin. 2002

189 citations recorded by the source corpus.

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 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 cohomology H • (X, C ) of any smooth symplectic resolution X↠V/G (multiplicative McKay correspondence). We prove further that if G⊂GL( h ) is an irreducible Weyl group and V= h ⊕ h ∗ , then no smooth symplectic resolution of V/G exists unless G is of types A , B , C .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fbcad88710fa2a80b18687ea55e53368e5a82ea

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In Geometry and Topology

1. I. Gordon
2. N. Hitchin
3. Y. Namikawa
4. M. Gualtieri
5. I. Losev
6. D. Waldram
7. G. Cavalcanti
8. M. Graña

In Mathematical Physics

1. I. Gordon
2. N. Hitchin
3. Y. Namikawa
4. M. Gualtieri
5. I. Losev
6. G. Cavalcanti
7. Stephen Griffeth
8. D. Waldram

Citations in the corpus

• Deformations of holomorphic Poisson manifolds (2011, background): “Poisson deformations of symplectic affine varieties have been studied in [7], [13], where unobstructedness results are proved as is the linear variation of the periods.”
• Symplectic reflection algebras (2007, background): “There are two key results in [69] for us.”
• On deformations of ℚ-factorial symplectic varieties (2005, background): “ty X. When X has non-isolated singularity, the usual deformation theory of Z (and X) has an infinite dimensional nature and does not work well. Instead, we should consider a “Poisson deformation” (cf. [G-K]). 2 Proof of Main Theorem (i) Q-factoriality: A normal variety Z is called Q-factorial if, for”
• Flops and Poisson deformations of symplectic varieties (2005, methodology): “on π: X˜ →Xwith π−1(U) ∼=U, ωextends to a regular 2-form on X˜. In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects T1 X and T 1 Y are not finite ”
• Poisson deformations of affine symplectic varieties (2006, background): “d to be smooth. But the arguments can be applied to our case. 7 be the evaluation map at t ∈Ad. We define a period map p : Ad →H2(X,C) by p(t) = evt(s). By the construction, p is a holomorphic map. In [G-K], Proposition (5.4), one can find another approach to the definition of the period map. The perio”