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

Abstract

In this paper, we prove a $\partial \bar{\partial }$-type lemma on compact Kähler 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 Kähler manifolds. Furthermore, we establish the Kähler version of an injectivity theorem originally formulated by F. Ambro in the algebraic setting. Notably, while O. Fujino previously addressed the Kähler case, our proof takes a different approach by avoiding the reliance on mixed Hodge structures for cohomology with compact support.

Final version, 52 pages, accepted for publication in Mathematische Annalen. Major revisions compared with v2: following the referee's suggestions, we add an alternative analytic proof of the Kähler Katzarkov-Kontsevich-Pantev unobstructed deformation theorem, i.e., Theorem A (see Appendix I, joint work with Sheng Rao). Comments are very welcome!