## December Workshop

### Speakers

- Andrea Cattaneo (Firenze)
- Dino Festi (Mainz)
- Ariyan Javanpeykar (Mainz)
- Bartosz Naskręcki (Poznan)
- Slawomir Rams (Jagiellonian University Krakow)
- Ulrike Rieß (ETH Zürich)
- Davide Cesare Veniani (Stuttgart)
- Federico Venturelli (Padova)

### Program

- Monday, December 9, 2019
- 10:00-11:00 2BC30 Ariyan Javanpeykar: Arithmetic and algebraic hyperbolicity
- 11:30-12:30 2BC30 Dino Festi: The Picard lattice of a K3 surface coming from Physics
- 14:00-15:00 2AB40 Slawomir Rams: On rational curves on polarized K3 surfaces
- 15:30-16:30 2AB40 Andrea Cattaneo: Some finiteness property of the automorphism group of a IHS manifolds
- 17:00-18:00 2AB40 Davide Veniani: Free involutions on ihs manifolds
- Tuesday, December 10, 2019
- 10:00-11:00 2BC30 Federico Venturelli: Triviality of the Alexander polynomial for a certain class of non-symmetric line arrangements
- 11:30-12:30 2BC30 Ulrike Rieß: Base loci of big and nef line bundles on irreducible symplectic varieties
- 14:00-15:00 2AB45 Bartosz Naskręcki: Applications of Shioda-Inose structures in arithmetic

The talks are in the Torre Archimede.

Coffee breaks and lunches are in room 701. We will go for dinner on Monday evening, if you want to participate please contact one of the organizers.

### Registration

In case you are interested in attending this workshop please write an email to one of the organizers.### Abstracts

**
Andrea Cattaneo **(Firenze)

*Some finiteness property of the automorphism group of a IHS manifolds*Abstract: Irreducible holomorphic symplectic manifolds can be considered as higher dimensional analogues of K3 surfaces, and in fact they enjoy many of the properties which hold for K3 surfaces. In our talk we focus on finiteness properties for the automorphism group, and we will show in particular that the automorphism group of an IHS manifold is finitely presented. Studying the structure of such group (or, more precisely, to the group of holomorphic and anti-holomorphic automorphisms) we can show that it has also a finite number of conjugacy classes of finite subgroups, and this fact has two interesting properties: the first is that the number of actions on an IHS manifold by finite groups is finite, and the second is that IHS manifolds has only a finite number of real forms.

**Dino Festi** (Mainz)

Title:*The Picard lattice of a K3 surface coming from Physics*

Abstract: Given the advancing precision of measurements carried out in modern Quantum Electrodynamics, equally precise theoretical predictions are required.
It turns out that the rationality problem for hypersurfaces plays essential role in the calculations needed to obtain such predictions.
In this talk we report on the study of a K3 surface arising in this context.
In particular, we will explain the techniques used to compute the Picard lattice of this particular surface.
This is joint work with Marco Besier (Mainz), Michael Harrison (MAGMA), Bartosz Naskrecki (Poznan).

**Ariyan Javanpeykar** (Mainz)

Title: *Arithmetic and algebraic hyperbolicity*

Abstract: What properties should algebraic varieties with only finitely many rational points in any given number field have? By Lang's conjecture, such a variety should have only finitely many symmetries (or dynamical systems). In this talk I will present recent work with Junyi Xie in which we verify this prediction: If X is a variety over a finitely generated field K of characteristic zero such that for every finite extension L/K the set of L-rational points of X is finite, then X has only finitely many rational dominant self-maps. If time permits, I will also explain present results in the "function field" setting.

**Bartosz Naskręcki** (Poznan)

Title: *Applications of Shioda-Inose structures in arithmetic*

Abstract: In this talk I will explain certain constructions related to motives in the decomposition of the H^2 cohomology of K3 surfaces of large Picard rank. One application is to motives arising from hypergeometric differential equations, another to motives which arise from a certain Picard-Fuchs equation of degree 4 and if time will permit to classical and Hilbert modularity of K3 surfaces.

**Slawomir Rams** (Krakow)

Title: * On rational curves on polarized K3 surfaces*

Abstract:
Rational curves play a fundamental role in the study of geometry of K3 surfaces.
In this talk, I will discuss some old and some new results on number and configurations of degree-d rational curves on polarized K3 surfaces.
This is joint work with M. Schuett (Hannover).

**Ulrieke Rieß ** (ETH Zürich)

Title:* Base loci of big and nef line bundles on irreducible symplectic varieties*

Abstract: In the first part of this talk, I give a complete description of the divisorial part of the base locus of big and nef line bundles on irreducible symplectic varieties (under certain conditions). This is a generalization of well-known results of Mayer and Saint-Donat for K3 surfaces. In the second part, I will present what is currently known on the non-divisorial part.

**Davide Veniani** (Stuttgart)

Title: *Free involutions on ihs manifolds*

Abstract: Irreducible holomorphic symplectic manifolds are one of the building blocks of kähler manifolds with vanishing first Chern class. In dimension 2 they are called K3 surfaces. Free involutions on K3 surfaces are quite interesting because they connect this class of surfaces with another class, namely Enriques surfaces. I will talk about a formula for the number of free involutions on a K3 surface (joint work with I. Shimada), the classification of K3 surfaces without any free involution (joint work with S. Brandhorst and S. Sonel) and the generalization to higher dimensions (joint work with S. Boissière).

**Federico Venturelli** (Padova)

Title: *Triviality of the Alexander polynomial for a certain class of non-symmetric line arrangements*

Abstract: The Alexander polynomial of a plane curve C is the characteristic polynomial \Delta_C of the algebraic monodromy action on H^1(F,C), where F is the Milnor fibre of C; although there exists a formula for the Alexander polynomial (involving type and relative position of the singularities of the curve), when C is a line arrangement one would like to determine \Delta_C based only on the combinatorial structure of C. In this talk, I will exhibit a class of `non-symmetric' line arrangements with trivial Alexander polynomial; this provides positive evidence for a conjecture of Papadima and Suciu. Key for the proof is a Lefschetz-type result on hyperplane sections which generalises the classical statement.