S ep 2 00 9 Poisson deformations of affine symplectic varieties Yoshinori Namikawa

A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the regular locus Xreg of X such that, for any resolution f : X̃ → X with f (Xreg) ∼= Xreg, the 2-form ω extends to a regular closed 2-form on X̃ (cf. [Be]). There is a natural Poisson structure { , } on X determined by ω. Then we can introduce the notion of a Poisson deformation of (X, { , }). A Poisson deformation is a deformation of the pair of X itself and the Poisson structure on it. When X is not a complete variety, the usual deformation theory does not work in general because the tangent object TX may possibly have infinite dimension. On the other hand, Poisson deformations work very well in many important cases where X is not a complete variety. Denote by PDX the Poisson deformation functor of a symplectic variety (cf. §1). In this paper, we shall study the Poisson deformation of an affine symplectic variety. The main result is: