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

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.