The logarithmic leaf complex and foliated d-semistability

TL;DR

This paper investigates holomorphic foliations on degenerating varieties using logarithmic structures, identifying obstructions to stability and developing a deformation theory with a versal moduli space.

Contribution

It introduces a logarithmic deformation framework for foliations on semistable degenerations, revealing obstructions and constructing a versal moduli space.

Findings

Identified local and global obstructions to d-semistability.

Developed a logarithmic deformation theory for foliations.

Established the existence of a versal hull for the moduli functor.

Abstract

We study holomorphic foliations on normal crossings varieties arising as semistable degenerations. We do so by we exploring the notion of foliated d-semistability using the language of logarithmic structures in the sense of Fontaine-Illusie. First, we identify both local and global obstructions to d-semistability. In order to analyze the existence of smoothings, we develop a logarithmic deformation theory of foliations and show that the corresponding moduli functor admits a versal hull.