Twistors, Hyper-Kaehler Manifolds, and Complex Moduli

TL;DR

This paper explores the global structure of complex moduli spaces, showing that they can have regions with unbounded local dimension, with examples derived from twistor theory of hyper-Kaehler manifolds.

Contribution

It provides explicit examples of moduli spaces with unbounded local dimension, highlighting limitations of the local finiteness in complex structure moduli.

Findings

Moduli spaces can contain regions with infinite local dimension.

Examples are constructed from twistor theory of hyper-Kaehler manifolds.

Local finiteness of moduli spaces does not hold globally.

Abstract

A theorem of Kuranishi tells us that the moduli space of complex structures on any smooth compact manifold is always locally a finite-dimensional space. Globally, however, this is simply not true; we display examples in which the moduli space contains a sequence of regions for which the local dimension tends to infinity. These examples naturally arise from the twistor theory of hyper-Kaehler manifolds.