Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie

Abstract

Using E. Witten's method, we prove asymptotic Morse inequalities for the d''-cohomology of tensor powers of a hermitian line bundle over a compact complex manifold: the H^q-dimension is bounded above by an integral of the (1,1)-curvature form, extended to the set of points of index q. The proof rests upon a spectral theorem which describes the asymptotic distribution of the spectrum of the Schrodinger operator associated to a large magnetic field. As an application, we find new geometric characterizations of Moisezon spaces, which improve Y. T. Siu's recent solution of the Grauert-Riemenschneider conjecture.1