## UniversitÃ© libre de Bruxelles, 13/05/2014

#### Bo Berndtsson

“Convexity of the Mabuchi K-energy and applications.”

The Mabuchi K-energy is a functional defined on the space of all Kahler metrics in a given cohomology class. A crucial property of the K-energy is that its critical points are metrics of constant scalar curvature. It has been known for a long time that the K-energy is convex along smooth geodesics. This fact is very important as a motivational guide, but its practical usefulness is limited since points in the space cannot be joined by a smooth geodesic. We will prove that the K-energy is in fact also convex along generalized geodesics and discuss applications of this to uniqueness problems. (This is joint work with Robert Berman.)

#### Jean-Pierre Demailly

“Rationally connected manifolds and semipositivity of the Ricci curvature.”

The talk will explain a structure theorem for compact Kähler manifolds with semipositive anticanonical bundle. Up to finite étale cover, it is shown that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the

non-existence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient is the characterization of uniruled nonsingular varieties by the property that the anticanonical bundle is not pseudoeffective (this is joint work with Frédéric Campana and Thomas Peternell).

#### David Witt-NystrÃ¶m

“Homogeneous Monge-Ampere equations and canonical tubular neighbourhoods in Kahler geoemtry”

In this talk I will describe some joint work with Julius Ross. By solving the Homogeneous Monge-Ampere equation on the deformation to the normal cone of a complex submanifold of a Kahler manifold, we get a canonical tubular neighbourhood adapted to both the holomorphic and the symplectic structure. If time permits I will describe an application, namely an optimal regularity result for certain naturally defined plurisubharmonic envelopes.