Abstract:The aim of this talk is to present an analogue in CR geometry of Demailly's holomorphic Morse inequalities. More precisely, we shall present asymptotic inequalities for the Kohn-Rossi cohomology groups with coefficients in large powers of a line bundle. A potential application is an extension of the Grauert-Riemenschneider criterion to noncompact complex manifolds. This is part of a joint project with George Marinescu (University of Cologne, Germany).