Global logarithmic deformation theory

Abstract

A classical problem in algebraic geometry is to construct smooth algebraic varieties with prescribed properties. In the approach via smoothings, one first constructs a degenerate scheme with the prescribed properties, and then shows the existence of a smooth variety degenerating to this scheme. Logarithmic geometry has given important new impulses to the second step of this approach, which we explore in this book. Degenerations, in particular in the context of mirror symmetry, often enjoy similar formal properties as smooth morphisms once considered from the logarithmic perspective. Logarithmic deformation theory has therefore become an effective tool for the construction of smoothings and the transfer of properties between smooth nearby fibers and the singular special fiber. The strongest existence result for deformations in classical algebraic geometry is the Bogomolov--Tian--Todorov theorem for Calabi--Yau varieties. A logarithmic variant, once established, constructs log smooth deformations. However, the logarithmic Bogomolov--Tian--Todorov theorem has resisted efforts to its proof for a while. Finally, a method to prove it was discovered in 2019 by Chan, Leung, and Ma. In this book, we explore this new approach to the logarithmic Bogomolov--Tian--Todorov theorem. We prove several variants of the abstract unobstructedness theorem, some of which are new and stronger than previously known results. We investigate its application to the global deformation theory of log smooth and mildly log singular spaces, obtaining unobstructedness results for log Calabi--Yau spaces, some log Fano spaces, and line bundles. Special care is taken to allow sufficiently mild log singularities, including all log singularities that appear in the Gross--Siebert construction of toric log Calabi--Yau mirror pairs.