5 Flops and Poisson deformations of symplectic varieties Yoshinori Namikawa

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). As an application of the Poisson deformation theory, we shall prove: