Holomorphic Morse inequalities on manifolds with boundary

Let X be a compact complex 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 of the Dolbeault cohomology groups with values in Lk, in terms of the curvature of L. 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