Logarithmic $\bf{\partial\bar\partial}$-lemma and several geometric applications (with an Appendix joint with Sheng Rao)

TL;DR

This paper proves a logarithmic $ar{ ext{d}}ar{ ext{d}}$-lemma on compact Kähler manifolds, confirming a conjecture, and applies it to extend several important results in complex geometry, including spectral sequence degeneracy and deformation unobstructedness.

Contribution

It establishes a new $ar{ ext{d}}ar{ ext{d}}$-lemma for logarithmic forms on Kähler manifolds, confirming Wan's conjecture and providing alternative proofs for key geometric theorems.

Findings

Proved the $ar{ ext{d}}ar{ ext{d}}$-lemma for logarithmic forms on compact Kähler manifolds.

Extended degeneracy results of spectral sequences to the Kähler setting.

Demonstrated unobstructed deformations of certain log Calabi-Yau pairs.

Abstract

In this paper, we prove a $\partial\bar{\partial}$-type lemma on compact K\"ahler manifolds for logarithmic differential forms valued in the dual of a certain pseudo-effective line bundle, thereby confirming a conjecture proposed by X. Wan. We then derive several applications, including strengthened results by H. Esnault-E. Viehweg on the degeneracy of the spectral sequence at the $E_1$-stage for projective manifolds associated with the logarithmic de Rham complex, as well as by L. Katzarkov-M. Kontsevich-T. Pantev on the unobstructed locally trivial deformations of a projective generalized log Calabi-Yau pair with some weights, both of which are extended to the broader context of compact K\"ahler manifolds. Furthermore, we establish the K\"ahler version of an injectivity theorem originally formulated by F. Ambro in the algebraic setting. Notably, while O. Fujino previously addressed the K\"ahler case, our proof takes a different approach by avoiding the reliance on mixed Hodge structures for cohomology with compact support.