Period Map for Non-Compact Holomorphically Symplectic Manifolds

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{C}.$\end{document} We show (under some additional, but mild, assumptions on X) that the coarse deformation space of the pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left\langle {X,\Omega } \right\rangle $\end{document} exists and is smooth, finite-dimensional and naturally embedded into H2(X). In particular, for an algebraic holomorphic symplectic manifold X which satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^i (\mathcal{O}_X ) = 0$ \end{document} for all i > 0, the coarse moduli of formal deformations is isomorphic to Spec \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{C}\left[\kern-0.15em\left[ {t_1 , \ldots ,t_n } \right]\kern-0.15em\right],$ \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t_1 , \ldots t_n $\end{document} are coordinates in H2(X).