Ueda's lemma via uniform H\"ormander estimates for flat line bundles

TL;DR

This paper proves uniform H"ormander-type $L^2$-estimates for the $ar{ abla}$-operator on all nontrivial flat line bundles over compact K"ahler manifolds, extending Ueda's lemma to a $ar{ abla}$-context.

Contribution

It introduces a $ar{ abla}$-version of Ueda's lemma with uniform estimates for flat line bundles, generalizing previous results and including a partial extension to Ricci-flat manifolds.

Findings

Established uniform $L^2$-estimates for $ar{ abla}$ on flat line bundles.

Reproduced Ueda's lemma in the $ar{ abla}$-operator setting.

Extended results to $(p,0)$-forms on Ricci-flat manifolds.

Abstract

We establish H\"ormander-type $L^2$-estimates for the $\overline{\partial}$-operators that hold uniformly for all nontrivial flat holomorphic line bundles on compact K\"ahler manifolds. Our result can be regarded as a $\overline{\partial}$-version of Ueda's lemma on the operator norm of \v{C}ech coboundaries for flat line bundles and indeed recovers the original version of Ueda's lemma for compact K\"ahler manifolds. A partial generalisation for $(p,0)$-forms on Ricci-flat manifolds is also given.