A Twisted Adiabatic Limit Approach to Vanishing Theorems for Complex Line Bundles

TL;DR

This paper introduces a new twisted adiabatic limit method to prove vanishing theorems for certain cohomology groups of complex line bundles on compact complex Hermitian manifolds, extending classical results beyond Kähler settings.

Contribution

It generalizes the twisted adiabatic limit approach to connections on complex line bundles, allowing vanishing theorems under broader geometric conditions without requiring Kähler structures.

Findings

Vanishing of specific D''-cohomology groups under positivity and smallness conditions.

Development of twisted Laplacians with comparison formulas similar to Bochner-Kodaira-Nakano identities.

Extension of vanishing theorems to non-Kähler manifolds and non-holomorphic line bundles.

Abstract

Given an $n$-dimensional compact complex Hermitian manifold $X$, a $C^\infty$ complex line bundle $L$ equipped with a connection $D$ whose $(0,\,1)$-component $D''$ squares to zero and a real-valued function $\eta$ on $X$, we prove that the $D''$-cohomology group of $L$ of any bidegree $(p,\,q)$ such that either $(p>q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\geq n+1)$ or $(p<q \hspace{1ex}\mbox{and}\hspace{1ex} p+q\leq n-1)$ vanishes when two extra hypotheses are made. The first hypothesis requires a certain real-valued, not necessarily closed, $(1,\,1)$-form depending on $p,\,q$, on the curvature of $D$ and on a $(1,\,1)$-form induced by $\eta$ to be positive definite. The second hypothesis requires the norm of $\partial\eta$ to be small relative to $|\eta|$. This theorem, for which we also give a number of variants, is proved by generalising our very recent twisted adiabatic limit construction for complex structures to connections on complex line bundles. This twisting of $D$ induces first-order differential operators acting on the $L$-valued forms, for which we obtain commutation relations involving their formal adjoints, and two twisted Laplacians for which we obtain a comparison formula reminiscent of the classical Bochner-Kodaira-Nakano identity. The main features of our results are that $X$ need not be K\"ahler, $L$ need not be holomorphic and the types of $C^\infty$ functions that $X$ supports play a key role in our hypotheses, thus capturing some of their links with the geometry of manifolds.