Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds

TL;DR

This paper investigates when pointwise isomorphic deformation families of compact complex manifolds are locally isomorphic, revealing conditions that differ between holomorphic and differentiable cases through a geometric analysis of the Kuranishi space.

Contribution

It provides a sufficient condition for local isomorphism in holomorphic families and demonstrates its insufficiency in differentiable families, including classification of counterexamples.

Findings

Sufficient condition established for holomorphic families

Counterexamples show the condition fails in differentiable families

Geometric analysis of the Kuranishi space underpins results

Abstract

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb R^p$, for some $p>0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold.