Vanishing Theorems on Covering Manifolds

Let M be an oriented even-dimensional Riemannian manifold on which a discrete group Γ of orientation-preserving isometries acts freely, so that the quotientX = M/Γ is compact. We prove a vanishing theorem for a half-kernel of a Γ-invariant Dirac operator on a Γ-equivariant Clifford module overM , twisted by a sufficiently large power of a Γ-equivariant line bundle, whose curvature is non-degenerate at any point of M . This generalizes our previous vanishing theorems for Dirac operators on a compact manifold. In particular, if M is an almost complex manifold we prove a vanishing theorem for the halfkernel of a spin Dirac operator, twisted by a line bundle with curvature of a mixed sign. In this case we also relax the assumption of non-degeneracy of the curvature. When M is a complex manifold our results imply analogues of Kodaira and Andreotti-Grauert vanishing theorems for covering manifolds. As another application, we show that semiclassically the spin quantization of an almost complex covering manifold gives an “honest” Hilbert space. This generalizes a result of Borthwick and Uribe, who considered quantization of compact manifolds. Application of our results to homogeneous manifolds of a real semisimple Lie group leads to new proofs of Griffiths-Schmidt and Atiyah-Schmidt vanishing theorems.