Université libre de Bruxelles, 13/05/2014
“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.)
“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).
“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.