Analytic Methods in Birational Geometry

The speakers for the sixth session, Analytic Methods in Birational Geometry, are:

  • Sébastien Boucksom (CNRS – CMLS École Polytechnique): “Valuations and singularities of plurisubharmonic functions

AbstractPlurisubharmonic (psh) functions can be viewed as a complex analytic analogue of convex functions, but they exhibit a much broader range of possible singularities, including those induced by ideals of holomorphic functions. Understanding psh singularities is a key issue in several questions of complex geometry, as illustrated by recent progress on the Yau-Tian-Donaldson conjecture. A powerful tool to do so is the classical notion of Lelong numbers and the theory of multiplier ideals, which open the way to a valuative approach that studies the analogue of psh functions on a (non-Archimedean) space of valuations. This lecture will present a gentle introduction to this circle of ideas.

  • Eleonora Di Nezza (CMLS École Polytechnique): “Pluripotential theory : how to get singular Kähler-Einstein metrics

Abstract: In the last 50 years pluripotential theory has played a central role in order to solve geometric problems, such as the existence of special metrics (e.g. Kähler-Einstein, csck) on a compact Kähler manifold. In this talk I am going to present some recent developments in pluripotential theory. These new tools are so flexible that they allow to study “singular” settings : we will then be able to work with a singular variety and/or to search for singular metrics. The talk is based on a series of joint papers with Támas Darvas and Chinh Lu.
(Talk accessible for researchers in geometry).

  • Andreas Höring (Université Côte d’Azur): “Stein complements in projective manifolds

Abstract: Let X be a complex projective manifold, and let Y be a prime divisor in X. If Y is ample, it is well-known that the complement X\Y is an affine variety. Vice versa, assume that X\Y is affine, or more generally a Stein manifold. Then X\Y does not contain any curve, in particular Y has positive intersection with every curve. This leads to our main question: if X\Y is Stein, what can we say about the normal bundle of Y? After some general considerations I will focus on the case where Y is the projectivised tangent bundle of some manifold M, and X is a “canonical extension”. We will see that the Stein property leads to many restrictions on the birational geometry of M. This is work in progress with Thomas Peternell.

  • Jian Song (Rutgers University): “Moduli space of Kähler-Einstein metrics of negative scalar curvature

Abstract:  Let K(n, V) be the space of n-dimensional compact Kähler-Einstein manifolds with negative scalar curvature and volume bounded above by V. We prove that any sequence in K(n, V) converges in pointed Gromov-Hausdorff topology to a finite union of complete Kähler-Einstein metric spaces without loss of volume, which is biholomorphic to an algebraic semi-log canonical model with its non-log terminal locus removed. We further show that the Weil-Petersson metric extends uniquely to a Kähler current with continuous local potentials on the KSB compactification of the moduli space of canonically polarized manifolds. In particular, the Weil-Petersson volume of the KSB moduli space is finite.

The schedule of the conference is the following:

  • Release of the videos: June 23, 2021 (YouTube)
  • Discussion: from June 23 to July 7, 2021 (Gitter)
  • Coffee break: July 7, 2021, at 17.00 CET (Zoom)