Titles and abstracts

▪ Gergely Bérczi (University of Oxford)

GIT for graded unipotent groups and jet differentials

Abstract: The orbit space for unipotent group actions on projective varieties is often too complicated to define a well-behaved quotient. In most applications, however, the acting group H is not unipotent but it contains a C^* which normalises the maximal unipotent subgroup U of H and acts with positive weights on the Lie algebra of U. We call these groups positively graded. After a short review of Mumford's GIT for reductive groups I will explain how the key geometric and computational features of GIT can be extended to positively graded groups. I will then demonstrate through jet reparametrisation groups how the topology of non-reductive moduli spaces can be recovered and I will explain new iterated residue formulae for intersection numbers turning up in classical problems such as Thom polynomials of singularities, the Green-Griffiths-Lang hyperbolicity conjecture and enumerative geometry problems of counting hypersurfaces with prescribed singularities in an ample linear system over a projective variety. I will explain how the interaction of these formulae gives a partial answer to a positivity conjecture of Rimanyi.

▪ Damian Brotbek (Université de Strasbourg)

On the hyperbolicity of general hypersurfaces

Abstract: The aim of this talk is to present a proof of a conjecture of Kobayashi stating that a general hypersurface of sufficiently high degree in projective space is hyperbolic. This proof is based on two main ingredients: A suitable Wronskian construction on the Demailly-Semple jet spaces. The construction of jet differential equations on deformations of Fermat type hypersurfaces, generalizing to higher order jet spaces a recent joint work with Darondeau.

▪ Frédéric Campana (Université de Lorraine, Nancy)

Positive foliations and fibrations with (orbifold) rationally connected fibres

Abstract: Foliations with positive minimal slope relatively to a movable class on a complex projective manifold X are shown to correspond exactly to fibrations with rationally connected fibres, extending results by Miyaoka and Bogomolov-McQuillan. The proof covers the case of smooth "orbifold pairs" (X,D) as well, once the notions of tangent bundle, foliation, rational connectedness, and "rational quotient" are suitably defined in this broader context. This is joint work with M. Păun, relying on previous joint work with T. Peternell.

▪ Junyan Cao (Université de Paris VI)

Kodaira dimension of algebraic fiber spaces over abelian varieties or surfaces

Abstract: We will report on a joint work with M. Păun. By using mainly the recent work of M. Păun and S. Takayama about the positivity of relatively canonical bundles, we give a proof of the Iitaka conjecture for algebraic fiber spaces over abelian varieties or over surfaces.

▪ Lionel Darondeau (IMPAN, Warsaw)

Complete intersection varieties with ample cotangent bundles

Abstract: This is a joint work with Damian Brotbek. We prove that a smooth projective variety contains many subvarieties with ample cotangent bundles, of each dimension up to half its own dimension. We obtain such subvarieties as certain complete intersections.

▪ Jean-Pierre Demailly (Institut Fourier, Université Grenoble Alpes)

Extension of holomorphic functions defined on non reduced analytic subvarieties

Abstract: The goal of the talk is to investigate L² extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results come with precise L² estimates and (probably) optimal curvature conditions.

▪ Ya Deng (Institut Fourier, Université Grenoble Alpes)

Effectivity in the Kobayashi and Debarre conjectures

Abstract: The goal of the talk is to deal with effective problems in the Kobayashi and Debarre conjectures, based on the work by Damian Brotbek and Lionel Darondeau. An important step for both problems is to obtain an effective estimate in the "almost" Nakamaye Theorem for the universal Grassmannian. We also show that if the line bundle L separates k-jets, the k-th Wronskian ideal sheaf for L is attained, which is enough to give an effective bound for the Kobayashi conjecture.

▪ Stéphane Druel (Institut Fourier, Université Grenoble Alpes)

A decomposition theorem for 5-dimensional singular spaces with trivial canonical class

Abstract: Building on recent results by Greb, Kebekus and Peternell, we prove an analogue of the Beauville-Bogomolov decomposition theorem for complex projective varieties with at worst klt singularities, of dimension at most 5.

▪ Philippe Eyssidieux (Institut Fourier, Université Grenoble Alpes)

Viscosity solutions of the Kähler-Ricci flow and the Song-Tian canonical metric

Abstract: Studying the long-term behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. The purpose of the talk is to explain a viscosity approach for degenerate complex Monge-Ampère flows.

▪ Julien Keller (Institut de Mathématiques de Marseille)

Finite dimensional approach to the Laplacian

Let (E,h) be a holomorphic Hermitian vector bundle over a polarized smooth complex manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian using an algebraic construction. If time allows, we will discuss some applications.

▪ Nefton Pali (Université de Paris-Sud Orsay)

On the concavity of Perelman's entropy functional

Abstract: I will explain a concavity result for Perelman's entropy functional over a smooth neighborhood of a Kähler-Ricci soliton inside the space of anti-canonically polarised complex structures over a Fano manifold. This result provides a gradient flow picture usefull for the solution of the existence problem for Kähler-Ricci solitons over Fano manifolds. The solution of the latter implies the general solution of the existence problem for Kähler-Einstein metrics over Fano manifolds with arbitrary automorphism group.

▪ Thomas Peternell (Universität Bayreuth)

Nef line bundles on Calabi-Yau varieties

Abstract: I will discuss nef line bundles L on Calabi-Yau manifolds, in particular in dimension 3. The basic problem is to show that L is semiample, i.e., some multiple is generated by global sections. I will report on joint recent work with V. Lazić and K. Oguiso on this abundance type problem.

▪ Dan Popovici (Institut de Mathématiques de Toulouse)

SKT Metrics and Frölicher Spectral Sequence

Abstract: With every Hermitian metric on an arbitrary compact complex manifold X, we associate a pseudo-differential operator acting on the smooth forms of any bidegree (p,q) whose kernel is shown to be isomorphic to the space of type (p,q) featuring at the second step in the Frölicher spectral sequence of X. Using this Hodge isomorphism, we prove the degeneration at E₂ of the Frölicher spectral sequence of any compact complex manifold supporting an SKT metric (i.e. such that ) whose torsion is small in a precise sense.

▪ Erwan Rousseau (Institut de Mathématiques de Marseille)

Curves on product-quotient surfaces

Abstract: Product-quotient surfaces are quotients of products of compact Riemann surfaces of genus at least two by a finite group. They provide interesting examples of surfaces of general type intensively studied by Bauer, Catanese, Pignatelli… We will explain some recent results on the geometry of rational and entire curves on these surfaces (joint work with J. Grivaux and J. Restrepo).

▪ Behrouz Taji (Albert Ludwigs University of Freiburg)

A remark on the Campana-Viehweg hyperbolicity conjecture

Abstract: A generalized conjecture of Campana and Viehweg predicts that the log Kodaira dimension of the base of a smooth family of canonically-polarized manifolds is bounded from below by the variation of the family. The case where the variation is maximal has been settled by the celebrated results of Viehweg, Zuo, Campana and Păun. Following similar methods, I will discuss the proof of the remaining part of this conjecture.

▪ Zheng Tao (Chinese Academy of Sciences, Beijing)

TBA

▪ Valentino Tosatti (Northwestern University)

Monge-Ampère equations for (n – 1,n – 1) forms

Abstract: I will discuss recent progress made by B.Weinkove and myself on solving complex Monge-Ampère type equations for (n – 1,n – 1) forms on compact complex manifolds, following a strategy suggested by Fu-Wang-Wu and Demailly. One of the applications of this theory is a proof of a conjecture of Gauduchon from 1984 (joint with B. Weinkove and G. Szekelyhidi).

▪ Jian Xiao (Institut Fourier, Université Grenoble Alpes)

A refined structure of the movable cone of curves

Abstract: We show how to obtain a refined structure of the movable cone of curves by Legendre-Fenchel type transforms, and explain its picture in convex geometry. This talk is based on joint work with Brian Lehmann.