Geometric & Functional Analysis GAFA

, Volume 12, Issue 6, pp 1265–1295 | Cite as

Period Map for Non-Compact Holomorphically Symplectic Manifolds

  • D. Kaledin
  • M. Verbitsky
Original Paper


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).


Symplectic Form Symplectic Manifold Formal Ring Deformation Space Formal Deformation 
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Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  1. 1.Steklov InstitueMoscowRussia
  2. 2.Independent UniversityMoscowRussia
  3. 3.Glasgow UniversityGlasgowScotland

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