Holomorphic Symplectic Geometry Ii

Hyperkähler embeddings and holomorphic symplectic geometry II. 0. Introduction. This is a second part of the treatment of complex analytic subvarieties of a holomorphically symplectic Kähler manifold. For the convenience of the reader, in the first two sections of this paper we recall the definitions and results of the first part ([V-pt I]). 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 closed analytic subvariety S of a holomorphically symplectic M , one can restrict the holomorphic symplectic form of M to the Zarisky tangent sheaf of S. If this restriction is non-degenerate outside of singularities of S and the same is true for the set of singular points of S, this sub-variety is called non-degenerately symplectic (Definition 2.2). Of course, non-degenerately symplectic subvarieties are of even complex dimension. The hyperkähler manifold is endowed with the canonical SU (2)-action in this cohomology space. Fix an induced complex structure on a hyperkähler manifold M. Let N be a closed analytic subset of M. Then N defines a cycle [N ] in cohomology of M. Denote the Poincare dual cocycle by N. In [V-pt I] we proved that if N is SU (2)-invariant then N is non-degenerately symplectic (Theorem 2.2 of [V-pt I]). Take a generic element M 0 in a given deformation class of compact holomorphically symplectic Kähler manifolds. Then all integer (p, p)-cycles on M 0 (i. e., all elements of H 2p (M, Z) ∩ H p,p (M)) are SU (2)-invariant (Proposition 2.2). According to Theorem 2.2 of [V-pt I], this immediately implies the following statement: All closed analytic subvarieties of N are non-degenerately symplectic. If such subvariety is smooth, it is also a hy-perkähler manifold (Proposition 2.1). Let M be a hyperkähler manifold with three complex structures I, J and K. The closed subset X ∈ M is called tri-analytic if X is analytic with respect to I, J and K. In this paper we prove the following. Let N be a closed analytic subset 1