Kuranishi and Teichm\"uller

TL;DR

This paper explores the local structure of the Teichmüller stack near Kähler points, showing an affirmative answer to a specific question in the Kähler case and indicating richer geometry in non-Kähler cases.

Contribution

It characterizes the local structure of the Teichmüller stack at Kähler points and discusses the complexity in non-Kähler cases.

Findings

At a generic Kähler point, the Catanese Kur=Teich question has an affirmative answer.

The local structure of the Teichmüller stack is described near Kähler points.

Non-Kähler manifolds exhibit a more complex Teichmüller geometry.

Abstract

The goal of this short article is to describe the local structure of the Teichm\"uller stack of [8] in the neighborhood of a K\"ahler point. In particular we show that at a generic K\"ahler point X, Catanese Kur=Teich question, when interpretated at the level of stacks, has an affirmative answer. The situation may be much more complicated if X is non-K\"ahler suggesting that Teichm\"uller spaces/stacks of non-K\"ahler manifold has a much richer geometry.