Uniform $L^2$-estimates for flat nontrivial line bundles on compact complex manifolds

TL;DR

This paper extends uniform $L^2$-estimates for $ar{ ext{d}}$-equations from compact Kähler to general compact complex manifolds, using Dolbeault isomorphism and Ueda's lemma.

Contribution

It generalizes previous $L^2$-estimates for flat line bundles to a broader class of complex manifolds, providing a new proof technique.

Findings

$L^2$-estimates are valid on all compact complex manifolds

The proof uses Dolbeault isomorphism and Ueda's lemma

Extends previous results from Kähler to non-Kähler manifolds

Abstract

In this paper, we extend the uniform $L^2$-estimate of $\bar{\partial}$-equations for flat nontrivial line bundles, proved for compact K\"ahler manifolds in the previous work, to compact complex manifolds. In the proof, by tracing the Dolbeault isomorphism in detail, we derive the desired $L^2$-estimate directly from Ueda's lemma.