1998/05/27 by Maxim Braverman, Braverman, Maxim
Mathematics · #14F17 (secondary) #32L20 (primary) #58G10 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #math.AG #math.DG #msc:14F17 #msc:32L20 #msc:58G10
paper · pdf · doi:10.48550/arxiv.math/9805127
A mistake in Theorem 3.13 is corrected. Some othe misprints are removed
openalex publication_date 1998/05/27 · arxiv created 1998/09/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the 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. These results are generalization of a vanishing theorem of Borthwick and Uribe. As an application we obtain a new proof of the classical Andreotti-Grauert vanishing theorem for the cohomology of a compact complex manifold with values in the sheaf of holomorphic sections of a holomorphic vector bundle, twisted by a large power of a holomorphic line bundle with curvature of a mixed sign. As another application we calculate the sign of the index of a signature operator twisted by a large power of a line bundle.