## LECTURES

**Lecture 1: An introduction to Pluripotential Theory **

(speaker: Ahmed Zeriahi and Do Hoang Son)

**Abstract**: Pluripotential Theory is the study of the "fine properties" of plurisubharmonic functions on domains in C^n, as well as in complex manifolds. These functions appear naturally in Kähler geometry as potentials of singular Kähler metrics on Kähler manifolds and also as local weights for positive singular hermitian metrics on holomorphic line bundles. They play an important role in many problems in Kähler geometry (e.g. the Calabi conjecture on Kähler singular varieties, the existence of singular Kähler-Einstein metrics, etc...). All these problems boil down to solving degenerate complex Monge-Ampère equations. The main goal of this course is to give an elementary introduction to this theory as developed by E. Bedford and B.A. Taylor in the early eighties and extended to Kähler manifold later by many authors. We will first recall some elementary facts from logarithmic potential theory in the complex plane and the Riemann sphere focusing on the Dirichlet problem for the Laplace operator. Then we will introduce the complex Monge-Ampère operator acting on bounded plurisubharmonic functions on domains in C^n or complex manifolds. Finally we will show how to solve the Dirichlet Problem for degenerate complex Monge-Ampère equations in strictly pseudo-convex domains in C^n using the Peron method.

**Lecture 2: Foundation of analytic methods in algebraic geometry**

(speaker: Shin-ichi Matsumura)

**Abstract:** The purpose of this lecture is to give the Nadel vanishing theorem and its generalizations. I first introduce the basic notion of holomorphic vector bundles, hermitian metrics, connections, Chern curvatures, and give two proofs for the Kodaira vanishing theorem. One is based on the theory of harmonic integrals and the other is based on the L^2-method of dbar-equation. In this step, I shall explain a fundamental relation between the theory of several complex variables and algebraic geometry. Finally, I give a generalization of the Kodaira vanishing theorem by using singular hermitian metrics and its applications.

**Lecture 3: Introduction to Kähler Geometry**(speaker: Lu Hoang Chinh)

**Abstract**: The purpose of these lectures is to quickly introduce several basic notions in Kähler geometry. We start by defining and giving examples of Kähler manifolds from two points of view: complex geometry and Riemannian geometry. We then study curvature tensors on a compact Kähler manifold specifying in the Ricci curvature. In the last lecture we show how problems concerning existence of canonical Kähler metrics is reduced to Partial differential equation (the Monge-Ampère equation).

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**Lecture 4: Introduction to Nevanlinna theory and its relation with Diophantine approximation**(speaker: Le Giang)

**Abstract:** The purpose of this lecture is to give basic notions in Nevanlinna theory and its relation with Diophantine. We start by defining Nevanlinna characteristic T(r, ƒ) which measures the rate of growth of a meromorphic function. It has been originally discovered by Osgood and Vojta, there is a formal analogy between Nevanlinna theory in complex analysis and certain results in Diophantine approximation. This connection has motivated the development in both subjects. In this lecture, I describe this analogy.

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**Lecture 5: Invariant metrics on non compact Complex manifolds and applications ** (speaker: Hervé Gaussier)

**Abstract:** The aim of this lecture is to give an overview of invariants metrics in noncompact complex manifolds (essentially in bounded domains in the complex Euclidean space). We will first describe the Poincaré metric in the disc, explaining the complex intrinsic point of view. Then we will introduce the Bergman metric, the Carathéodory metric and the Kobayashi metric, generalizations of the Poincaré metric in complex manifolds. We will explain how these metrics may be used to study holomorphic extension of holomorphic maps in strongly pseudoconvex domains. Finally we will study the global behaviour of (real) geodesics of these metrics, in relation with the notion of Gromov hyperbolicity.