1998/09/24 by Maxim Braverman, Braverman, Maxim
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.AG #math.DG #math.RT
paper · pdf · doi:10.48550/arxiv.math/9809144
LaTeX 2e; 25 pages
arxiv created 1998/09/24 · openalex publication_date 1998/09/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be an oriented even-dimensional Riemannian manifold on which a discrete group Γ of orientation-preserving isometries acts freely, so that the quotient X=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 over M, 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 half-kernel of a \spinc 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 \spinc 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.