Deformations of trianalytic subvarieties of hyperk\"ahler manifolds

Abstract

Let $M$ be a compact complex manifold equipped with a hyperk\"ahler metric, and $X$ be a closed complex analytic subvariety of $M$. In alg-geom/9403006, we proved that $X$ is trianalytic, i. e., complex analytic with respect to all complex structures induced by the hyperk\"ahler structure, provided that $M$ is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of $X$ are trianalytic and naturally isomorphic to $X$ as complex analytic varieties. We show that this isomorphism is compatible with the metric induced from $M$. Also, we prove that the Douady space of complex analytic deformations of $X$ in $M$ is equipped with a natural hyperk\"ahler structure.