Inégalités de Morse Holomorphes Singulières

Abstract

We generalize Demailly’s holomorphic Morse inequalities to the case of a line bundle E equipped with a singular metric on an arbitrary compact complex manifold X. Our inequalities give an estimate of the cohomology groups with values in the tensor power E⊗k twisted by the corresponding sequence of multiplier ideal sheaves introduced by Nadel. The allowed singularities are of the following type: the metric is locally given by a weight exp(−φ) where $$\phi \sim \tfrac{c}{2}\log (\Sigma |f_j |^2 )$$ with holomorphic fj. As a consequence, we obtain a necessary and sufficient analytic condition, invariant by bimeromorphism, for a manifold X to be Moishezon. This characterization improves a result given by Ji and Shiffman. We finally recall and improve some results of Kollar in order to show that the corresponding sufficient conditions obtained by Siu and Demailly in the smooth case are not necessary.