Hyperkähler embeddings and holomorphic symplectic geometry I. Mikhail Verbitsky,

Hyperkähler embeddings and holomorphic symplectic geometry I. 0. Introduction. In this paper we are studying complex analytic subvarieties of a given Kähler manifold which is endowed with a holomorphic symplectic structure. By Calabi-Yau theorem, the holomorphically symplectic Kähler mani-folds can be supplied with a Ricci-flat Riemannian metric. This implies that such manifolds are hyperkähler (Definition 1.1). Conversely, all hyperkähler manifolds are holomorphically symplectic (Proposition 2.1). For a given closed analytic subvariety S of a holomorphically symplectic M , one can restrict the holomorphic symplectic form of M to the Zarisky tangent sheaf to S. If this restriction is non-degenerate outside of singular-ities of S, this subvariety is called non-degenerately symplectic. (Definition 2.2). Of course, such subvarieties are of even complex dimension. Take a generic element N in a given deformation class of a holomorphi-cally symplectic Kähler manifolds. We are proving that all complex analytic subvarieties of N are non-degenerately symplectic (Theorem 2.3). In particular , all closed analytic subvarieties of N are of even complex dimension. If such subvariety is smooth, it is also a hyperkaehler manifold (Proposition 2.1). Contents. 1. Hyperkähler manifolds. 2. Holomorphic symplectic geometry. 3. The action of so(5) on the differential forms over a hyperkähler manifold.