Compact Hyperkaehler Manifolds: Basic Results

Abstract

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity of such manifolds and their ample resp. Kaehler cones. It is proved that the period map surjects any non-empty connected component of the moduli space of marked manifolds onto the corresponding period domain. We also establish an anlogue of the transitivity of the Weyl-action on the set of chambers of a K3 surface. As a converse of a higher-dimensional version of the `Main Lemma' of Burns and Rapoport we prove that two birational irreducible symplectic manifolds are deformation equivalent. This compares nicely with a result of Batyrev and Kontsevich.