Algebraic structures on hyperkähler manifolds

Let $M$ be a compact hyperkähler manifold. The hyperkähler structure equips $M$ with a set $\cal R$ of complex structures parametrized by $\Bbb C P^1$, called {\em the set of induced complex structures.} It was known previously that induced complex structures are {\it non-algebraic}, except maybe a countable set. We prove that a {\it countable} set of induced complex structures is {\it algebraic}, and this set is dense in $\cal R$. A more general version of this theorem was proven by A. Fujiki.

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Published 1 January 1996