Poisson deformations of symplectic quotient singularities

Dmitry Kaledin; Victor Ginzburg

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Poisson deformations of symplectic quotient singularities

Victor Ginzburg, Dmitry Kaledin

Algebraic Geometry · Submitted 19 Dec 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.67 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.14 · mentions code or data availability 0.02.

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arXiv listing abstract v5. Results and the main claim are underlined; show every role.

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 \to 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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety.	contribution 0.68	0.92	0.10	asserted 1.00
s1	In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup.	background 0.93	0.12	0.02	unclear 0.58
s2	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.	result 0.99	0.92	0.96	asserted 0.98
s3	That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence).	result 0.80	0.92	0.08	asserted 0.99
s4	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.	result 1.00	0.93	0.06	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original 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 \to 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.

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3 arXiv papers cite this in 5 sentences

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Papers citing it · 4 sentences without a confident label

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By [ref] , the second Poisson cohomology $^2(_V^n^S_n+1)$ is zero unless $V = 2$, in which case it is one-dimensional.

Poisson traces for symmetric powers of symplectic varieties · Poisson deformations and zero-dimensional symplectic leaves · 0.89

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