Log Structures on Generalized Semi-Stable Varieties

TL;DR

This thesis develops a framework for semi-stable log structures on morphisms with mild singularities, providing conditions for their existence and uniqueness, thus advancing the understanding of singularities in algebraic geometry.

Contribution

It introduces an obstruction theory characterizing the existence of semi-stable log structures on a new class of morphisms with weaker singularities than normal crossings.

Findings

Obstruction vanishes iff semi-stable log structures exist

Semi-stable log structures are unique when no power is involved

Canonical structures are established on the family of morphisms

Abstract

This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms.