Class Of Compact Complex Manifolds Research Articles

Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics

This paper is intended to start a series of works aimed at proving that if in a (smooth) complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. A new method of attack, whose starting point originates in Demailly’s work, is introduced. While we hope to be able to address the general case in the near future, two important special cases are established here: the one where the Hodge numbers h0,1 of the fibres are supposed to be locally constant and the one where the limit fibre is assumed to be a strongly Gauduchon manifold. The latter is a rather weak metric assumption giving rise to a new, rather general, class of compact complex manifolds that we hereby introduce and whose relevance to this type of problems we underscore.

Einstein–Kähler metrics on certain complex bundles

We study a class of compact complex manifolds, with positive first Chern class, fibered over products of projective spaces. We prove that these bundles carry Einstein–Kähler metrics when the projective spaces of the basis have the same dimension. When this dimensional condition is not satisfied, we estimate their Ricci tensor in another paper.

On Ricci curvature of certain complex bundles

We study a class of compact complex manifolds, with positive first Chern class C 1 , fibered over products of projective spaces. When the spaces of the basis don't have the same dimension, we know that these bundles cannot carry Einstein–Kähler metrics. In this article, we construct a metric belonging to C 1 , with an optimal lower bound of the Ricci curvature.

Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature

To study C(0)a priori estimates for solutions to certain complex Monge-Ampère equations, I introduce a coherent sheaf of ideals and show that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem. This technique is used to establish the existence of Kähler-Einstein metrics of positive scalar curvature on a very large class of compact complex manifolds.