Algebraic Structures on Hyperkähler Manifolds Algebraic Structures on Hyperkähler Manifolds

Let M be a compact hyperkähler manifold. The hy-perkähler structure equips M with a set R of complex structures parametrized by CP 1 , called the set of induced complex structures. It was known previously that induced complex structures are non-algebraic, except may be a countable set. We prove that a countable set of induced complex structures is algebraic, and this set is dense in R. A more general version of this theorem was proven by A. Fujiki. The structure of this paper is following. In Subsection 1.1 we define hyperkähler manifolds and induced complex structures. In Subsection 1.2, we define induced complex structures of general type. The main result of this paper and its proof are given in Section 2.