Bergman kernels and local holomorphic Morse inequalities

Let (X,) be a hermitian manifold and let L^k be a high power of a hermitian holomorphic line bundle over X. Local versions of Demaillys holomorphic Morse inequalities (that give bounds on the dimension of the Dolbeault cohomology groups associated to L^k), are presented - after integration they give the usual holomorphic Morse inequalities. The local weak inequalities hold on any hermitian manifold (X,), regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in Mathematics Subject Classification (2000): 32A25, 32L10, 32L20in final form: 19 June 2003