Flops and Poisson deformations of symplectic varieties

By Y. Namikawa. 2005

77 citations recorded by the source corpus.

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 TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introduced a Poisson scheme and studied a Poisson deformation of it. A Poisson deformation is the deformation of the pair of a scheme itself and a Poisson structure on it. When X is a convex symplectic variety, X admits a natural Poisson structure induced from a symplectic 2-form ω; hence one can consider its Poisson deformations. Then they are controlled by the Poisson cohomology. The Poisson cohomology has been extensively studied by Fresse [Fr 1], [Fr 2]. In some good cases, it can be described by well-known topological data (Corollary 10). The first application of the Poisson deformation theory is the following two results:

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/42ef581fa3cc2b495919e4d121fd1a3ccfe8ccfc

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In Geometry and Topology

  1. I. Losev
  2. V. Ginzburg
  3. T. Arakawa
  4. A. Premet
  5. I. Gordon
  6. Ben Webster

In Mathematical Physics

  1. I. Losev
  2. V. Ginzburg
  3. T. Arakawa
  4. A. Premet
  5. I. Gordon
  6. Ben Webster

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