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A logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor

Xueyuan Wan

TL;DR

The paper addresses solving a logarithmic $\bar{\partial}$-equation on a compact Kähler manifold with a smooth divisor $D$ by employing a cyclic covering trick to transfer the problem to a cover where harmonic theory applies. It proves that for any $\alpha$ with $\bar{\partial}\partial\alpha=0$, there exists $x$ solving $\bar{\partial}x=\partial\alpha$, and then derives multiple powerful consequences in the logarithmic setting. These include the closedness of logarithmic forms, an injectivity theorem, $E_1$-degeneration of the logarithmic Frölicher spectral sequence, and unobstructed logarithmic deformations for $D\in|-2K_X|$. The results provide analytic foundations for degeneration and deformation theory in logarithmic geometry on Kähler manifolds and motivate a conjecture for extending to simple normal crossing divisors with weighted line bundles.

Abstract

In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at $E_1$, and we also prove that the pair $(X,D)$ has unobstructed deformations for any smooth divisor $D\in|-2K_X|$.

A logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor

TL;DR

The paper addresses solving a logarithmic -equation on a compact Kähler manifold with a smooth divisor by employing a cyclic covering trick to transfer the problem to a cover where harmonic theory applies. It proves that for any with , there exists solving , and then derives multiple powerful consequences in the logarithmic setting. These include the closedness of logarithmic forms, an injectivity theorem, -degeneration of the logarithmic Frölicher spectral sequence, and unobstructed logarithmic deformations for . The results provide analytic foundations for degeneration and deformation theory in logarithmic geometry on Kähler manifolds and motivate a conjecture for extending to simple normal crossing divisors with weighted line bundles.

Abstract

In this paper, we solve a logarithmic -equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at , and we also prove that the pair has unobstructed deformations for any smooth divisor .

Paper Structure

This paper contains 12 sections, 16 theorems, 104 equations.

Table of Contents

  1. Preliminaries
  2. Logarithmic connection
  3. Cyclic covering
  4. Logarithmic $\bar{\partial}$-equation
  5. Logarithmic $\bar{\partial}$-equation
  6. Solving the $\bar{\partial}$-equation
  7. Some applications
  8. Closedness of logarithmic forms
  9. An injectivity theorem
  10. Degeneration of spectral sequences
  11. Logarithmic deformation
  12. Further discussions

Key Result

Theorem 1

Let $X$ be a compact Kähler manifold and $D=\sum_{i=1}^rD_i$ be a smooth divisor in $X$, let $L$ be a holomorphic line bundle over $X$ with $L^N=\mathcal{O}_X(D)$. For any $\alpha\in A^{0,q}(X,\Omega^p_X(\log D)\otimes L^{-1})$ with $\bar{\partial}\partial\alpha=0$, the following equation has a solution $x\in A^{0,q-1}(X,\Omega^{p+1}_X(\log D)\otimes L^{-1})$.

Theorems & Definitions (30)

  • Theorem 1: =Theorem \ref{['thm1']}
  • Corollary 2: =Corollary \ref{['Cor1']}
  • Proposition 3: =Proposition \ref{['Cor2']}
  • Theorem 4: =Theorem \ref{['thm2']}
  • Theorem 5: =Theorem \ref{['thm4']}
  • Conjecture 6
  • Definition 1.1: Viehweg
  • Proposition 1.2
  • proof
  • Proposition 1.3: BHPV, Iacono
  • ...and 20 more