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

Runze Zhang

TL;DR

The paper proves a logarithmic $\partial\bar{\partial}$-lemma on compact Kähler manifolds for logarithmic forms valued in the dual of a pseudo-effective line bundle, confirming Wan's conjecture. It then derives applications including $E_1$-degeneration for the logarithmic de Rham complex, a Kähler version of Ambro’s injectivity theorem, and the unobstructed locally trivial deformations of generalized log Calabi–Yau pairs, all extended to compact Kähler settings. The authors present two complementary proofs: an algebraic approach via degeneration of a DGBVA and an analytic, Tian–Todorov–style power-series construction (the latter in Appendix with Sheng Rao). Collectively, the results broaden the Bogomolov–Tian–Todorov framework to logarithmic settings on Kähler manifolds and connect log Hodge theory with deformation theory and spectral-sequence degeneracy, with potential implications for mirror symmetry and nonprojective Calabi–Yau-type geometries.

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.

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

TL;DR

The paper proves a logarithmic -lemma on compact Kähler manifolds for logarithmic forms valued in the dual of a pseudo-effective line bundle, confirming Wan's conjecture. It then derives applications including -degeneration for the logarithmic de Rham complex, a Kähler version of Ambro’s injectivity theorem, and the unobstructed locally trivial deformations of generalized log Calabi–Yau pairs, all extended to compact Kähler settings. The authors present two complementary proofs: an algebraic approach via degeneration of a DGBVA and an analytic, Tian–Todorov–style power-series construction (the latter in Appendix with Sheng Rao). Collectively, the results broaden the Bogomolov–Tian–Todorov framework to logarithmic settings on Kähler manifolds and connect log Hodge theory with deformation theory and spectral-sequence degeneracy, with potential implications for mirror symmetry and nonprojective Calabi–Yau-type geometries.

Abstract

In this paper, we prove a -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 -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.

Paper Structure

This paper contains 23 sections, 34 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Backgrounds and main results
  3. Idea of the proofs
  4. Overview and outlook
  5. Logarithmic connection and logarithmic complex
  6. Warm-up: all $q_i>0$
  7. Hodge decomposition: conic version
  8. Proof of Theorem \ref{['Thm 1.3']}
  9. General case: not all $q_i>0$
  10. de Rham--Kodaira decomposition for conic currents
  11. Log conic current
  12. Proof of Theorem \ref{['n,q-main']}
  13. Applications
  14. Closedness of twisted logarithmic forms
  15. Degeneracy of spectral sequence
  16. ...and 8 more sections

Key Result

Theorem A

Let $X$ be an $n$-dimensional compact Kähler manifold, and let $D=\sum_{i=1}^s D_i$ be a simple normal crossing divisor on $X$. Assume that there exist rational weights $\{a_i\}_{1\le i\le s}\subset [0,1]\cap \mathbb{Q}$ such that which means that for some positive integer $N$. Then, the locally trivial (infinitesimal) deformations of the pair $(X,D)$ are unobstructed. That is, for such deformat

Theorems & Definitions (100)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem B: Wan18
  • Remark 1.3
  • Remark 1.4
  • Theorem C
  • Theorem D
  • Remark 1.5
  • Definition 2.1: EV92
  • ...and 90 more