Period Map for Non-Compact Holomorphically Symplectic Manifolds

Abstract

We study the deformations of a holomorphic symplectic manifold X, not necessarily compact, over a formal ring. We always assume both X and the symplectic form Ω to be algebraic over $\mathbb{C}.$ We show (under some additional, but mild, assumptions on X) that the coarse deformation space of the pair $\left\langle {X,\Omega } \right\rangle $ exists and is smooth, finite-dimensional and naturally embedded into H2(X). In particular, for an algebraic holomorphic symplectic manifold X which satisfies $H^i (\mathcal{O}_X ) = 0$ for all i > 0, the coarse moduli of formal deformations is isomorphic to Spec $\mathbb{C}\left[\kern-0.15em\left[ {t_1 , \ldots ,t_n } \right]\kern-0.15em\right],$ where $t_1 , \ldots t_n $ are coordinates in H2(X).