# Program

# Schedule

### Monday, May 13

- 9:00-10:30 == Corvaja
- 10:30-11:00 == Coffee Break - PK-5675
- 11:00-12:00 == Javanpeykar
- 12:00-14:30 == Lunch
- 14:30-15:30 == Grieve
- 15:30-16:00 == Coffee Break - PK-5675
- 16:00-17:00 == Buium

### Tuesday, May 14

- 9:00-10:30 == Javanpeykar
- 10:30-11:00 == Coffee Break - PK-5675
- 11:00-12:00 == Corvaja
- 12:00-14:30 == Lunch
- 14:30-15:30 == Brotbek
- 15:30-16:00 == Coffee Break - PK-5675
- 16:00-17:00 == Zuo

### Wednesday, May 15

- 9:00-10:00 == Levin
- 10:00-11:00 == Ascher
- 11:00-11:30 == Coffee Break - PK-5675
- 11:30-12:30 == Turchet
- Free afternoon

### Thursday, May 16

- 9:00-10:30 == Javanpeykar
- 10:30-11:00 == Coffee Break - PK-5675
- 11:00-12:00 == Corvaja
- 12:00-14:30 == Lunch
- 14:30-15:30 == Heier
- 15:30-16:00 == Coffee Break - PK-5675
- 16:00-17:00 == Chen (special colloquium)
- 17:00-21:00 == Wine and Cheese and Colloquium dinner

### Friday, May 17

- 9:00-10:30 == Corvaja
- 10:30-11:00 == Coffee Break - PK-5675
- 11:00-12:00 == Riedl
- 12:00-14:30 == Lunch
- 14:30-15:00 == Sun
- 15:00-15:30 == Phung
- 15:30-16:00 == Coffee Break - PK-5675
- 16:00-17:00 == Noguchi

# Titles and Abstracts

### Hyperbolicity of varieties of log general type

*Speaker*: Kenneth B. Ascher

*Abstract*: This talk will survey some recent results regarding hyperbolicity of varieties of log general type. There are several classical results showing that positivity of the cotangent bundle implies various notions of hyperbolicity for projective varieties coming from algebraic, arithmetic, and differential geometry. The goal of this talk is to review these results and discuss recent work which generalizes these results for log pairs / quasi-projective varieties. This is joint work with K. DeVleming and A. Turchet.

### On the positivity of the logarithmic cotangent bundle

*Speaker*: Damian Brotbek

*Abstract*: This is a joint work with Ya Deng. Given a smooth logarithmic pair (X,D), the positivity of its logarithmic cotangent bundle has strong implications on the geometric properties of the complement of D in X, in particular on its hyperbolicity properties. As soon as the dimension of X is greater than one and that D is not empty, the logarithmic cotangent bundle of (X,D) cannot be ample. In this talk, we describe the obstruction to ampleness and provide examples for which the logarithmic cotangent bundle is “as ample as possible”.

### Transcendental numbers as solutions to arithmetic differential equations

*Speaker*: Alexandru Buium

*Abstract*: Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental functions are solutions to algebraic differential equations; in this talk we show that, in a similar way, some remarkable p-adic transcendental numbers (including certain “p-adic periods”) are solutions to arithmetic differential equations. Following a clue from a paper of Manin, we then speculate on the possibility of understanding the algebraic relations among periods via Galois groups of arithmetic differential equations.

### Introduction to birational classification theory in dimension three and higher

*Speaker*: Jungkai Chen

*Abstract*: One of the main themes of algebraic geometry is to classify algebraic varieties and to study various geometric properties of each of the interesting classes. Classical theories of curves and surfaces give a beautiful framework of classification theory. Recent developments provide more details in the case of dimension three. We are going to introduce the three-dimensional story and share some expectations for even higher dimensions.

### Integral and rational points on algebraic varieties

*Speaker*: Pietro Corvaja

*Abstract*: One of the main goals of Diophantine geometry is linking the geometric properties of an algebraic varieties with the distribution of the integral or rational points on it. Broad conjectures by Lang, Vojta and Campana predicts the degeneracy (resp., the potential density) of the integral or rational points on varieties of hyperbolic (resp. special) type.

We shall discuss these conjectures by working out in details some concrete cases. Especially, degeneracy results based on Diophantine approximation techniques will be proved.

### Stability, complexity of rational points and arithmetic of linear series

*Speaker*: Nathan Grieve

*Abstract*:
I will survey concepts that are near to K-stability and which have origins in toric geometry. A main goal will be to explain their role in measuring arithmetic complexity of rational points, for example questions in Diophantine approximation for projective varieties. There are also important connections to measures of growth and positivity of line bundles. These arithmetic results build on a number of earlier related works including those of Ru-Vojta, McKinnon-Roth, Evertse-Ferretti and Fujita.

### A generalized Schmidt subspace theorem for closed subschemes

*Speaker*: Gordon Heier

*Abstract*:
In this talk, we will present a generalized version of Schmidt’s subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. Our proof builds on previous work by Evertse and Ferretti, Corvaja and Zannier, and others, and uses standard techniques from algebraic geometry such as notions of positivity, blowing-ups and direct image sheaves. As an application, we recover a higher-dimensional Diophantine approximation theorem of K. F. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof, while simultaneously extending the scope of the use of Seshadri constants in this context in a natural way. We will also discuss recent progress regarding applications to the degeneracy of integral points on varieties in the complement of divisors. This is joint work with Aaron Levin.

### Hyperbolicities: arithmetic, algebraic, and non-archimedean

*Speaker*: Ariyan Javanpeykar

*Abstract*:
Brody hyperbolic projective varieties over the complex numbers are extremely rich in properties. For instance, such varieties have only finitely many automorphisms, and every surjective endomorphism is actually an automorphism of finite order. Consequently, if one believes the conjectures of Green-Griffiths and Lang, these properties should be shared by varieties which are “arithmetically” hyperbolic or “algebraically” hyperbolic (in the sense of Demailly). In these lecture series, we will see how to establish some of these properties, and thereby verify many of the predictions made by the conjectures of Green-Griffiths and Lang.

### Greatest common divisors in Diophantine approximation and Nevanlinna theory

*Speaker*: Aaron Levin

*Abstract*: In 2003, Bugeaud, Corvaja, and Zannier gave an upper bound for the greatest common divisor gcd(a^{n}-1,b^{n}-1), where *a* and *b* are fixed integers and n varies over the positive integers. In contrast to the elementary statement of their result, the proof required deep results from Diophantine approximation. I will discuss a higher-dimensional generalization of their result and some recent related results (with Julie Wang) in Nevanlinna theory.

### Some relations of the value distribution theory and the Diophantine geometry at proof level

*Speaker*: Junjiro Noguchi

*Abstract*: We begin with a proof of Raynaud’s Theorem (1983) for torsion points of semi-abelian varieties by making use of a Big Picard Theorem
(N. 1981). It is interesting to see the “o-minimal structure” to bridge those tow theories. In the same spirit we will discuss some arithmetic properties of possibly transcendental holomorphic sections in semi-abelian schemes (joint with Corvaja and Zannier). We will then find some new problems.

### Integral points on abelian varieties over function fields

*Speaker*: Xuan Kien Phung

*Abstract*: In this short talk, I will discuss some results developing the ideas of Parshin in 1990 using hyperbolic and homotopy methods to study integral points on abelian varieties over one dimensional complex function fields.

### Ampleness of the cotangent bundle of complete intersections

*Speaker*: Eric Riedl

*Abstract*: If X is a smooth variety with ample cotangent bundle, then X will be hyperbolic. Thus, ampleness of the cotangent bundle is a strong hyperbolicity property. Brotbek-Darondeau and Xie prove that for X a general n-dimensional complete intersection in projective space of codimension c, where c is at least n, will have ample cotangent bundle provided that the degrees of the defining equations are sufficiently large. We investigate whether we can obtain better bounds on the degrees if we let c become larger with respect to n. We prove that this is indeed the case, proving a polynomial bound on the degree in terms of n and c, and showing that this bound converges to 3 when c becomes large relative to n. This is joint work with Izzet Coskun.

### On the Borel hyperbolicity of moduli space of polarized varieties

*Speaker*: Ruiran Sun

*Abstract*: In this short talk, I will discuss the Borel hyperbolicity of the base space of a family of polarized varieties. In particular, I’ll present how to use the Viehweg-Zuo’s construction and an old extension theorem of Steven Lu to show the Borel hyperbolicity in some special case. Some known results and supporting evidence will also be presented.

### Geometric Vojta conjecture for non-isotrivial ramified covers of G_{m}^{2}

*Conférencier*: Amos Turchet

*Résumé*: We present a generalization of a result of Corvaja and Zannier for weak algebraic hyperbolicity of fibered threefolds whose generic fiber is a ramified cover of G_{m}^{2}. This builds on an explicit bound for the number of multiple zeroes of a polynomial with non-constant coefficients evaluated at S-units. This is joint work with Laura Capuano.

### Higgs bundle over base of log Family and hyperbolicity

*Speaker*: Kang Zuo

*Abstract*: Given a family of log varieties over a base Y with the degeneration locus S. Assuming the bigness of the determinant bundle of the direct image of the power of the relative log dualizing sheaf, I shall outline Viehweg-Zuo’s construction of the logarithmic Higgs bundle over (Y,S) arising from the deformation theory, showing that the kernel of Kodaira-Spencer map is negatively curved.

Different types of hyperbolicity on Y\S should follow from the negativity of the kernel. In particular, Y\S should be Borel hyperbolic, namely, analytic maps from algebraic curves into Y\S are algebraic.

If the iterated Kodaira-Spence map along the analytic map has maximal length, this fact follows directly from an old theorem of Steven Lu. In his talk, Ruiran Sun will give detailed discussion on the approach to this conjecture in a joint project of Lu-Sun-Zuo.