Analytic semi-universal deformations in logarithmic complex geometry

TL;DR

This paper extends classical deformation theory to compact complex analytic spaces with logarithmic structures, establishing the existence of semi-universal deformations using a two-step process.

Contribution

It generalizes the existence of semi-universal deformations to the logarithmic setting, building on Douady's classical approach.

Findings

Existence of semi-universal deformations for spaces with logarithmic structures

Extension of classical deformation results to a broader logarithmic context

Application of a two-step process involving infinite and finite-dimensional constructions

Abstract

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the '70. We follow Douady's two steps process approach consisting of an infinite-dimensional construction of the semi-universal deformation space followed by a finite-dimensional reduction.