Holomorphic Morse inequalities on manifolds with boundary
  Let X be a compact manifold with boundary and let L^k 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 L^k. 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. 
Robert Berman