2004/02/06 by Robert Berman, Robert D. Berman, Berman, Robert · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #math.CV
paper · pdf · doi:10.48550/arxiv.math/0402104
42 pages. Typos fixed. Section on Strong Morse inequalites added. Treatment of model case simplified by use of new metric, conformally equivalent to the old one
openalex publication_date 2004/02/06 · arxiv created 2004/12/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a compact manifold with boundary and let Lk be a high power of a hermitian holomorphic line bundle over X. When X has no boundary, Demailly's holomorphic Morse inequalities give asymptotic bounds on the dimensions on the Dolbeault cohomology groups with values in Lk. We extend Demailly's inequalities to the case when X has a boundary by adding a boundary term expressed as a certain average of the curvature of the line bundle and the Levi curvature of the boundary. Examples are given that show that the inequalities are sharp and it is shown that they are compatible with hole filling. The most interesting case is when X is a pseudoconcave manifold with a positive line bundle L. If the curvature of L is conformally equivalent to the Levi curvature of the boundary, the Morse inequalities are shown to be equalities, so that the space of global sections of L have maximal asymptotic growth, i.e. L is big. The sharp examples show that the corresponding cohomology group of (0,1)-forms may have maximal asymptotic growth, as well, unless the conformal equivalence holds.