Geometry of logarithmic forms and deformations of complex structures

TL;DR

This paper introduces a new approach to solving $ar{ ext{d}}$-equations for logarithmic forms, generalizes key theorems in logarithmic Hodge theory, and proves the unobstructedness of deformations of log Calabi-Yau structures using differential geometry.

Contribution

It develops a $ar{ ext{d}}$-lemma for logarithmic forms, extends deformation theory of log Calabi-Yau structures, and provides geometric proofs of classical theorems.

Findings

A new method for solving $ar{ ext{d}}$-equations for logarithmic forms.

Generalization of Deligne's closedness and degeneracy theorems.

Proof of unobstructedness of deformations of log Calabi-Yau pairs.

Abstract

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at $E_1$-level, as well as certain injectivity theorem on compact Kahler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kahler manifolds, we construct the extension for any logarithmic $(n,q)$-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by differential geometric method.