Practical

This session will be our first two-day event! It is scheduled from Monday to Tuesday October 6-7 2025 at the IECL, University of Lorraine (Nancy),

The address is:

Salle de Conférences
Institut Élie Cartan de Lorraine - University of Lorraine (Nancy)
Faculté des Sciences et technologie Campus,
Boulevard des Aiguillettes, BP 70239
54506 Vandœuvre-lès-Nancy

How to get there

Schedule

Monday 6 October:

12:30–14:00	Lunch (for those who have already arrived)
14:00–15:30	Stefan Kebekus: Hyperbolicity in C-pairs
15:30–16:00	coffee break
16:00–17:00	Benoît Cadorel: Hyperbolicity of projective hypersurfaces via Green-Griffiths jet differentials
Around 19:30	Dinner

Tuesday 7 October:

09:30–10:30	Aryaman Patel: The Hitchin morphism and the Chen-Ngô conjecture
10:30–11:00	coffee break
11:00–12:00	Wendelin Lutz: The Morrison Cone Conjecture under deformation
12:00–	Lunch (for those who haven’t left)

Speakers, titles and abstracts

Stefan Kebekus: Hyperbolicity in C-pairs

Almost twenty years ago, Campana introduced C-pairs to complex geometry. Interpolating between compact and non-compact geometry, C-pairs capture the notion of “fractional positivity” in the “fractional logarithmic tangent bundle”. Today, they are an indispensible tool in the study of hyperbolicity, higher-dimensional birational geometry and several branches of arithmetic geometry. This talk reports on joint work with Erwan Rousseau. We clarify the notion of a “morphism of C-pairs”, define (and prove the existence of) a “C-Albanese variety”, and discuss the beginnings of a Nevanlinna theory for “orbifold entire curves”.

Benoît Cadorel: Hyperbolicity of projective hypersurfaces via Green-Griffiths jet differentials

A complex projective variety X is said to be (Brody) hyperbolic if it does not admit any entire curve, i.e. any non-constant holomorphic map starting from the complex line. The Kobayashi conjecture asserts that a generic hypersurface of high degree should be Brody hyperbolic; it is even expected that the bound on the degree should be linear in the dimension of the hypersurface. The first complete proof of this conjecture was only given by Brotbek in 2017, for hypersurfaces of very large degree. It has become clear in the last few years that an efficient way of proving hyperbolicity results for hypersurfaces is to apply jet differential techniques, via the so-called strategy of “slanted vector fields”, initiated by Siu: this strategy has been refined across the years by many authors (Diverio-Merker-Rousseau, Darondeau, Brotbek, Deng, Demailly, Riedl-Yang…). Until quite recently, the best known bounds it provided were at least exponential in the dimension.

A core part of this strategy of slanted vector fields consists in picking an adequate jet space sitting above the hypersurface, before studying the base locus of a natural tautological line bundle defined on it. In the last decade, Bérczi-Kirwan have managed to construct a new jet space via techniques of non-reductive GIT, which allowed them to spectacularly improve the previously known bounds, to a polynomial in the dimension. Actually, another candidate for this choice of jet spaces has existed for quite a long time, but seems to have been a bit overlooked in this problem: the famous Green-Griffiths jet space, studied by these two authors around 1980. As we will explain, this latter space can also be used in the strategy above, and, surprisingly enough, it again gives a polynomial bound.

Aryaman Patel: The Hitchin morphism and the Chen-Ngô conjecture

Let X be a smooth projective variety over algebraically closed k of characteristic zero. The Hitchin morphism is a map from the moduli stack of Higgs bundles to the so-called Hitchin base, which sends a Higgs bundle to the characteristic polynomial of the Higgs field. This map is surjective when X is a curve, but for dim X>1, it is in general not surjective. The natural question is- can we describe the image of the Hitchin morphism in the latter case? Chen-Ngô showed that the Hitchin morphism factors through a closed subscheme of the Hitchin base, called the “spectral base”, which can be described explicitly. They conjecture that the spectral base is the image of the Hitchin morphism. We prove a stronger version of this conjecture when X satisfies some conditions.

Wendelin Lutz: The Morrison Cone Conjecture under deformation

Let Y be a Calabi—Yau variety. The Morrison Cone Conjecture is a fundamental conjecture in Algebraic Geometry on the geometry of the nef cone and the movable cone of Y: while these cones are usually not rational polyhedral, the cone conjecture predicts that the action of Aut(Y) on Nef(Y) admits a rational polyhedral fundamental domain, and that the action of Bir(Y) on Mov(Y) admits a rational polyhedral fundamental domain.

Even though the conjecture has been settled in special cases, it is still wide open in dimension at least 3. We prove that if the cone conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.