Subvarieties in non-compact hyperkähler manifolds

Let $M$ be a hyperkähler manifold, not necessarily compact, and $S\cong \C P^1$ the set of complex structures induced by the quaternionic action. Trianalytic subvariety of $M$ is a subvariety which is complex analytic with respect to all $I \in \C P^1$. We show that for all $I \in S$ outside of a countable set, all compact complex subvarieties $Z \subset (M,I)$ are trianalytic. For $M$ compact, this result was proven in \cite{_Verbitsky:Symplectic_II_} using Hodge theory.

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Published 1 January 2004