Non-algebraic hyperkähler manifolds

Abstract

We study the algebraic dimension a(X) of a compact hyperkäahler manifold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kähler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0, n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results for “generalised hyperkähler” manifolds and studies nef lines bundles. © 2010 Journal of Differential Geometry.