Workshop on Complex Hyperbolicity

February 22nd and 23rd, 2024

ESAGA, Essen

Speakers

Aim

The aim of the workshop is to study recent developments in the theory of complex hyperbolic manifolds. Special focus will be laid upon hodge-theoretic aspects and algebraicity/finiteness results. Among other things we plan to learn about the contents of the following papers

* Damian Brotbek, Yohan Brunebarbe: Arakelov-Nevanlinna inequalities for variations of Hodge structures and applications, arXiv:2007.12957

* Ariyan Javanpeykar, Robert Kucharczyk: Algebraicity of analytic maps to a hyperbolic variety, DOI:10.1002/mana.201900098

* Ariyan Javanpeykar, Steven Lu, Ruiran Sun, Kang Zuo: Finiteness of pointed maps to moduli spaces of polarized varieties, arXiv:2310.06784

For more details, see the subsequent section.

Abstracts

Damian Brotbek: Arakelov-Nevanlinna inequalities

In this series of talks I will present some results concerning Nevanlinna theory for variations of Hodge structure, and related questions.
The question is the following: Suppose one is given a pair (X,D) where W is a projective manifold and D is a simple normal crossing divisor, and that U=X\D supports a variation of Hodge structure. Under suitable assumption on the period mapping, it is known since the work of Griffiths that U is Brody hyperbolic. It is also known that (under similar assumptions), the pair (X,D) is of general type. In view of Griffiths fundamental conjecture on entire curves, it is a reasonable question to study the distributions of entire curves in X with respect to the boundary divisor D. The purpose of this talk is to explain how to obtain a Second Main Theorem in this setting, which is an analogue in the transcendental setting of the so-called Arakelov inequality in the algebraic case. This is based on a joint work with Brunebarbe.

Talk 1 : Algebraic hyperbolicity and Nevanlinna theory for parabolic Riemann surfaces

In this introductory talk I will try to explain why much of Nevanlinna theory can be understood as a transcendental analogue of algebraic hyperbolicity. I will also explain how this can be formalized using parabolic Riemann surfaces. I will in particular mention the notion of Nevanlinna pair introduced by Yan He and Min Ru.

Talk 2 : Nevanlinna theory via the negativity of the sectional curvature

In this second talk, I will explain how one can use the existence of a metric with negative (bounded above) holomorphic sectional curvature on the open subset U=X\D to obtain a Second Main Theorem for the pair (X,D). This mainly relies on a version of the lemma of logarithmic derivative. As a consequence, I will prove that if U is hyperbolically embedded in X, then there exists a Second Main Theorem for the pair (X,D).

Talk 3 : The Arakelov-Nevanlinna inequality and applications

In this last talk I will explain how the previous results can be directly applied to the setting of variations of Hodge structures simply from the known curvature properties of the Griffiths-Schmidt metric. I will also give some applications of this result generalizing results of Nadel on the existence of hyperbolic coverings of X.

Ariyan Javanpeykar: Hyperbolicity vs finiteness

In these three talks I will present some new evidence for the conjectures of Lang, Vojta, Green-Griffiths and Campana relating hyperbolicity and finiteness properties of algebraic varieties. We will start with an overview talk (Talk 1) which will be followed with two talks containing novel results. In our second talk we will explain how Shafarevich's conjecture for the moduli space of curves extends to moduli spaces of higher-dimensional varieties (by work of Viehweg-Zuo and Campana-Paun). In the third talk I will move to a different aspect of these conjectures provided by Campana's conjectures on special varieties.

Talk 1. "The Lang-Vojta conjectures"

I will present the Lang-Vojta conjectures for complex algebraic varieties. Namely, for a projective variety X, the following are equivalent:

1) X is hyperbolic (i.e., X has no entire curves)

2) For every projective variety Y, the moduli space of maps Hom(Y,X) is a projective variety (i.e,. pi_0(Hom(Y,X)) is finite) (i.e., X is bounded)

3) Every subvariety of X is of general type

4) X has only finitely many rational points (i.e., X is Mordellic)

5) and more...

I will also explain how this conjecture extends to the quasi-projective setting (where counterexamples can be given to a naive formulation).

Talk 2. "Moduli spaces"

What would be a good class of varieties to test the Lang-Vojta conjectures? I will explain that moduli spaces of canonically polarized smooth projective varieties are hyperbolic (Viehweg-Zuo), satisfy boundedness (Viehweg-Zuo), and that every subvariety is of general type (Campana-Paun). The story here goes back to Shafarevich who conjectured at the 1962 ICM in Stockholm related statements for the moduli space of curves (his conjecture was proven by Arakelov and Parshin in the 70s). In this talk I will explain these results and how we can prove finiteness results for rational points over function fields (joint work with Steven Lu, Ruiran Sun, and Kang Zuo) by means of a novel rigidity result for pointed maps.

Talk 3. "Campana's conjectures"

To understand what a hyperbolic variety is, it is also crucial to understand the properties of varieties *very far* from being hyperbolic. Campana introduced the class of special varieties for this reason and formulated a precise conjecture similar to the Lang-Vojta conjecture. In the early nineties, his conjecture was discovered to contradict another conjecture suggested by Abramovich and Colliot-Thélène in a paper of Harris-Tschinkel. This conflict has not been resolved yet, as we do not understand the arithmetic of Campana's one-dimensional orbifold pairs (as opposed to the arithmetic of algebraic curves, where we have Faltings's theorem). In this talk, I will present novel evidence (joint works with Finn Bartsch+Erwan Rousseau and Finn Bartsch+Olivier Wittenberg+Frederic Campana) that should convince you that Campana's conjectures are the correct ones.

Schedule

THURSDAY

10:00 - 10:30 Welcome and Coffee/Tea

10:30 - 11:30 Brotbek 1
11:45 - 12:45 Javanpeykar 1
Lunch break
14:30 - 15:30 Javanpeykar 2
Coffee break
16:00 - 17:00 Winkelmann

FRIDAY

10:00 - 11:00 Brotbek 2
11:30 - 12:30 Javanpeykar 3
Lunch break
14:30 - 15:30 Brotbek 3

Location

ESAGA, University of Duisburg-Essen, Essen, Seminar Room WSC-N-U-3.05.

A plan showing how to find the building (which is not located on the main campus) is here.

Organisers

Daniel Greb (UDE)

Christian Lehn (RUB)

Funding

ANR-DFG Project "QuaSiDy"

DFG-RTG 2553 "Symmetries and Classifying Spaces - Analytic, arithmetic, and derived"