Wirtinger numbers and holomorphic symplectic immersions

Abstract

For any subvariety of a compact holomorphic symplectic Kaehler manifold, we define the number W(X), which we call Wirtinger number. We show that $W(X)\le 1$, and the equality is reached if and only if the subvariety $X\subset M$ is trianalytic, i. e. compactible with the hyperkaehler structure on M. For a sequence $X_1 \arrow X_2 \arrow ... X_n\arrow M$ of immersions of simple holomorphic symplectic manifolds, we show that $W(X_1)\le W(X_2)\le >...\le W(X_n)$.

minor corrections - September 1999; 10 pages