In\'egalit\'es de Morse et vari\'et\'es de Moishezon

Abstract

The central topic of this thesis is the study of some properties of a class of complex compact manifolds~: Moishezon manifolds. In the first part, we generalize J.-P. Demailly's holomorphic Morse inequalities to the case of a line bundle equipped with a metric with analytic singularities on an arbitrary compact complex manifold. Our inequalities give an estimate of the cohomology groups with values in the line bundle tensor powers twisted by the corresponding sequence of multiplier ideal sheaves introduced by Nadel. As a consequence, we obtain a necessary and sufficient analytic condition, invariant by bimeromorphism, for a manifold to be Moishezon. In the second part, we use Mori theory to analyze the structure of Moishezon manifolds with infinite cyclic Picard group, with big canonical bundle, and which become projective after one single blow-up with smooth center. We study the dimension and the structure of the center of the blow-up. In dimension four, we show that this locus is always a surface, and when the canonical bundle