Ueda's lemma via uniform Hörmander estimates for flat line bundles

Abstract

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

Minor modifications, details added