WEEKLY ACTIVITIES

Time: 14h Monday, April 3, 2017.
Location: Room 3, Building A14, Institute of Mathematics.
Abstract: A fundamental problem in Kahler Geometry is to find a canonical metric on a compact Kahler manifold X with "nice properties" that reflect the topological and geometric properties of the manifold. This has been made precise by E. Calabi in the early fifthies in an ambitious program. He stated a famous conjecture and asked for the existence of various kinds of "canonical metrics" on a given compact Kahler manifold.
The goal of this lecture are:

• To explain the Calabi conjecture as well as the problem of the existence of Kahler-Einstein metrics on compact Kahler manifolds and show how these problems boil down to solving complex Monge-Ampere equations.
• To state the important theorems by Calabi-Yau (1978) and Aubin-Yau (1978) who solved these problems when the given compact Kahler manifold has zero first Chern class and negative first Chern class respectively.
• To explain how to extend these results to the case of normal complex projective varieties with "mild singularities".
• In this last case, the corresponding complex Monge-Ampere equations are degenerate. We will show how Pluripotential Theory provides soft methods to prove existence and uniqueness of Singular Kahler-Einstein metrics on Calabi-Yau varieties and normal complex projective varieties of general type.