Symplectic singularities from the Poisson point of view
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D Kaledin
Abstract
Introduction
In symplectic geometry, it is often useful to consider the so-called Poisson bracket on the algebra of functions on a C∞ symplectic manifold M. The bracket determines, and is determined by, the symplectic form; however, many of the features of symplectic geometry are more conveniently described in terms of the Poisson bracket. When one turns to the study of symplectic manifolds in the holomorphic or algebro-geometric setting, one expects the Poisson bracket to be even more useful because of the following observation: the bracket is a purely algebraic structure, and it generalizes immediately to singular algebraic varieties and complex-analytic spaces.
The appropriate notion of singularities for symplectic algebraic varieties has been introduced recently by A. Beauville [A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), 541–549.] and studied by Y. Namikawa [Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), 597–623.], [Y. Namikawa, Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123–147.]. However, the theory of singular symplectic algebraic varieties is in its starting stages; in particular, to the best of our knowledge, the Poisson methods have not been used yet. This is the goal of the present paper.
© Walter de Gruyter
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- Exterior products of zero-cycles
- Harmonic analysis on totally disconnected groups and irregularities of point distributions
- Relative cyclic homology of square zero extensions
- Weighted Fano threefold hypersurfaces
- Surfaces expanding by the inverse Gauß curvature flow
- Symplectic singularities from the Poisson point of view
- Multiplicative rule in the Grothendieck cohomology of a flag variety
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Articles in the same Issue
- Exterior products of zero-cycles
- Harmonic analysis on totally disconnected groups and irregularities of point distributions
- Relative cyclic homology of square zero extensions
- Weighted Fano threefold hypersurfaces
- Surfaces expanding by the inverse Gauß curvature flow
- Symplectic singularities from the Poisson point of view
- Multiplicative rule in the Grothendieck cohomology of a flag variety
- Generalized double affine Hecke algebras of higher rank
- Defining relations of the tame automorphism group of polynomial algebras in three variables
