# Workshop on Complex Hyperbolicity

# February 22nd and 23rd, 2024

## ESAGA, Essen

## Speakers

Damian Brotbek (Nancy)

Ariyan Javanpeykar (Nijmegen)

Jörg Winkelmann (Bochum)

## 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*, **a**rXiv: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

##### Talk 1. "The Lang-Vojta conjectures"

##### Talk 2. "Moduli spaces"

##### Talk 3. "Campana's conjectures"

## 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"