Log Smooth Deformation Theory

TL;DR

This paper develops a foundational framework for log smooth deformation theory, analyzing infinitesimal liftings and establishing a representable hull for the deformation functor, with applications to algebraic varieties and normal crossing varieties.

Contribution

It introduces a comprehensive foundation for log smooth deformation theory, including the existence of a representable hull and applications to various deformation types.

Findings

Established the existence of a representable hull for the log smooth deformation functor.

Generalized relative deformation theory for pairs of algebraic varieties and divisors.

Connected log smooth deformations with existing theories by Kawamata and Namikawa.

Abstract

This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.