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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

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).

Keywords

Symplectic Form Symplectic Manifold Formal Ring Deformation Space Formal Deformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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