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

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.