Deformation theory of singular symplectic n-folds

By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially a few examples of symplectic manifolds are discovered, it seems an important task to seek new symplectic manifolds by deforming symplectic varieties. In this paper we shall study a projective symplectic variety from a view point of deformation theory. If X has a resolution π : X̃ → X such that (X̃, πω) is a symplectic manifold, we say that X has a symplectic resolution. Our first results are concerned with a birational contraction map of a symplectic manifold.