Subvarieties in non-compact hyperkaehler manifolds

TL;DR

This paper proves that in non-compact hyperkaehler manifolds, most compact complex subvarieties are trianalytic, extending known results from the compact case and highlighting the structure of subvarieties across complex structures.

Contribution

It establishes that for non-compact hyperkaehler manifolds, outside a countable set, all compact complex subvarieties are trianalytic, generalizing previous compact manifold results.

Findings

Most compact complex subvarieties are trianalytic outside a countable set of complex structures.

The result extends known theorems from compact to non-compact hyperkaehler manifolds.

The proof relies on properties of the quaternionic action and complex structures.

Abstract

Let M be a hyperkaehler manifold, not necessarily compact, and $S\cong CP^1$ 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 $I \in CP^1$. 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 alg-geom/9403006 using Hodge theory.