Wirtinger numbers and holomorphic symplectic immersions

Abstract.

For any subvariety of a compact holomorphic symplectic Kähler manifold, we define the symplectic Wirtinger number W(X). We show that $W(X) \leqslant 1,$ and the equality is reached if and only if the subvariety $X \subset M$ is trianalytic, i.e. compatible with the hyperkähler structure on M. For a sequence $X_1 \to X_2 \to \ldots X_n \to M$ of immersions of simple holomorphic symplectic manifolds, we show that $W\left( {X_1 } \right) \leqslant W\left( {X_2 } \right) \leqslant \ldots \leqslant W\left( {X_n } \right).$