Extension theorems and Kahler-Einstein matrics

Abstract : This thesis consists in two parts: -In the first part, we first deal with a Kahler version of the famous Ohsawa-Takegoshi extension theorem; then, a problem of extending the closed positive currents. Our motivation comes from the Siu's conjecture on the invariance of plurigenera over a Kahler family. Indeed, in the proof of his famous theorem, the Ohsawa-Takegoshi theorem plays an important role. It is, therefore, natural to think that the proof for the conjecture involves an extension theorem of Ohsawa-Takegoshi type in the Kahler case. Because of the technical difficulties coming from the regularization process of quasi-psh functions over the compact Kahler manifolds, we only obtain two special cases of the hoped result. As for the extension of closed positive currents, our result is a special case of the conjecture which predicts that every closed positive current defined over the central fiber in a Kahler cohomology class twisted by the first Chern class of the canonical bundle admits an extension. -In the second part, we are interested in the uniqueness of the solutions of the equations of generalized Monge-Ampère type, a generalized Bando-Mabuchi theorem concerning the Kahler-Einstein metrics over Fano manifolds. We follow the method introduced by Berndtsson and generalize his result by working with a closed positive current in place of a klt pair in his context. The properties of the convexity of the Bergman metrics play an important role in this part
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Li Yi. Extension theorems and Kahler-Einstein matrics. Mathematics [math]. Université de Lorraine, 2012. English. ⟨NNT : 2012LORR0151⟩. ⟨tel-02074622⟩

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