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

Given an n-dimensional compact complex Hermitian manifold X, a C^ complex line bundle L equipped with a connection D whose (0, \, 1) -component D'' squares to zero and a real-valued function on X, we prove that the D''-cohomology group of L of any bidegree (p, \, q) such that either (p>q 1exand1ex p+q n+1) or (p<q 1exand1ex p+q 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 to be positive definite. The second hypothesis requires the norm of to be small relative to ||. 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^ functions that X supports play a key role in our hypotheses, thus capturing some of their links with the geometry of manifolds.