Any component of moduli of polarized hyperkaehler manifolds is dense in its deformation space

Let M be a compact hyperkaehler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in $H^2(M)$ defines a divisor $D_v$ in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W.Comment: 17 pages, 4 figures, v. 5.0, the introduction is cleaned up, a reference to [KV] adde

Hypercomplex manifolds with trivial canonical bundle and their holonomy

Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric. Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle. We show that the holonomy of Obata connection on M is contained in SL(n,H). In Appendix we apply these arguments to compact nilmanifolds equipped with abelian hypercomplex structures, showing that such manifolds have holonomy in SL(n,H).Comment: 14 pages, minor misprints correcte