Nonalgebraic hyperkähler manifolds

We study the algebraic dimension $a(X)$ of a compact hyperkähler 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.