[Paper Review] Poisson deformations of symplectic quotient singularities

Victor Ginzburg, D. Kaledin|arXiv (Cornell University)|Dec 19, 2002

Advanced Algebra and Geometry23 references4 citations

TL;DR

This paper establishes that the Calogero-Moser deformation provides a versal Poisson deformation for symplectic quotient singularities $V/G$, where $V$ is a symplectic vector space and $G \subset \mathrm{Sp}(V)$ is a finite group. It proves a multiplicative McKay correspondence, showing that the cohomology ring of any smooth symplectic resolution $X \to V/G$ is isomorphic to the associated graded algebra of the center of the group algebra $\mathbb{C}[G]$ with respect to a natural filtration, and shows that such resolutions do not exist for Weyl groups of types $D_n$, $E_n$, $F_4$, or $G_2$ unless $G$ is of type $A$, $B$, or $C$. The result provides a deep link between Poisson geometry, representation theory, and algebraic geometry.

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\subset 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 o 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.

Motivation & Objective

• To establish a connection between smooth symplectic resolutions and symplectic deformations of affine Poisson varieties.
• To determine the algebra structure on the cohomology of any smooth symplectic resolution $X \to V/G$ via a Poisson deformation framework.
• To resolve the existence question for symplectic resolutions of $V/G$ when $G$ is a Weyl group, showing non-existence for types $D_n$, $E_n$, $F_4$, $G_2$.
• To prove a multiplicative McKay correspondence identifying $H^\bullet(X, \mathbb{C})$ with the associated graded algebra of the center of $\mathbb{C}[G]$.

Proposed method

• Use of Poisson cohomology to analyze deformations of the Poisson structure on $V/G$, particularly $HP^2(V/G)$, which classifies infinitesimal Poisson deformations.
• Construction of the Calogero-Moser deformation as a universal Poisson deformation of $V/G$, leveraging the geometry of the quotient and the action of $G$.
• Definition of a filtration $F_k(\mathbb{C}[G])$ on the group algebra $\mathbb{C}[G]$ by the rank of $\mathrm{id} - g$, and the associated graded algebra $\mathrm{gr}^F(\mathbb{Z}G)$.
• Application of global Poisson deformation theory to relate the deformation space of $V/G$ to the cohomology of a resolution $X \to V/G$, using the fact that symplectic resolutions are crepant and the symplectic form lifts.
• Use of Hochschild cohomology and the Künneth formula to relate the cohomology of $X$ to the structure of $\mathrm{gr}^F(\mathbb{Z}G)$.
• Leveraging the theory of graded algebras and étale descent to establish an equivalence between categories of coherent sheaves on $X$ and the associated graded algebra, preserving tensor structures.

Experimental results

Research questions

• RQ1 Is the Calogero-Moser deformation a versal Poisson deformation of the symplectic quotient singularity $V/G$?
• RQ2 What is the algebra structure on the cohomology ring $H^\bullet(X, \mathbb{C})$ of a smooth symplectic resolution $X \to V/G$?
• RQ3 For which finite subgroups $G \subset \mathrm{Sp}(V)$ does a symplectic resolution of $V/G$ exist?
• RQ4 Can the cohomology of $X$ be described in terms of the representation theory of $G$, specifically via the center of $\mathbb{C}[G]$?
• RQ5 Does the multiplicative McKay correspondence hold, identifying $H^\bullet(X, \mathbb{C})$ with $\mathrm{gr}^F(\mathbb{Z}G)$?

Key findings

• The Calogero-Moser deformation provides a versal Poisson deformation of $V/G$, meaning it captures all possible infinitesimal Poisson deformations of the singularity.
• For $G$ a Weyl group acting on $V = \mathfrak{h} \otimes \mathbb{C}^2$, no smooth symplectic resolution exists unless $G$ is of type $A_n$, $B_n$, or $C_n$.
• The cohomology ring $H^\bullet(X, \mathbb{C})$ of any smooth symplectic resolution $X \to V/G$ is isomorphic as a graded algebra to $\mathrm{gr}^F(\mathbb{Z}G)$, the associated graded algebra of the center of $\mathbb{C}[G]$ with respect to the filtration by rank of $\mathrm{id} - g$. This is the multiplicative McKay correspondence.
• The resolution $X$ has no odd rational cohomology, as $H^{2k+1}(X, \mathbb{C}) = 0$ for all $k \geq 0$, due to the isomorphism with $\mathrm{gr}^F(\mathbb{Z}G)$, which is concentrated in even degrees.
• The dimension of $H^i(X, \mathbb{C})$ equals the dimension of the $i$-th graded piece of $\mathrm{gr}^F(\mathbb{Z}G)$, confirming a known dimension equality in a stronger, multiplicative form.
• The action of the Lie algebra of Poisson derivations on $HP^2(V/G)$ is trivial when $HP^1(V/G) = 0$, which is used to establish the versality of the Calogero-Moser deformation.