## Program

**Thursday, June 28th**

lunch from 13:00

13.50 – 14.00 opening

14.00 – 15.00 K. Hulek: *On the cohomology of toroidal compactifications of ${\cal A}_g$ *

15.15 – 16.15 B. Siebert: *Generalized Theta Functions*

coffee

16:45 - 17:15 A-D. Halanay: * Moduli spaces of vector bundles on principal elliptic fibrations*

17:30 - 18:30 A. Langer: *Vanishing theorems for nef line bundles*

**Friday, June 29th **

09.00 – 10.00 J. Ritter: *Equivariant Iwasawa Theory*

10.15 – 10.45 S. Arias-de-Reyna: *Compatible systems of symplectic Galois representations with large residual image*

coffee

11.15 – 12.15 E. Ullmo:

*The hyperbolic Ax-Lindemann conjecture for projectve Shimura varieties*

and applications to the André-Oort conjecture

and applications to the André-Oort conjecture

lunch

13:30 - 14:30 S. Petersen:

*Independence of $\ell$-adic Galois representations over function fields*

14:45 - 15:15 B. Naskręcki:

*Mordell-Weil ranks of families of elliptic curves associated to*

Pythagorean triples

Pythagorean triples

coffee

15:45 - 16:45 F. Brown:

*On the motivic fundamental group of ${\Bbb P}^1$ minus $3$ points*

**Saturday, June 30th**

09:00 - 10:00 A. Corti: *Algebraic geometry and $G_2$ manifolds*

10:15 - 10:45 N. Medeiros: *On the polar degree of projective hypersurfaces *

coffee

11:15 - 12:15 I. Bauer: *Inoue surfaces and Inoue type manifolds: weak rigidity*

lunch

13:30 - 14:30 M. Schütt: *Picard numbers of quintic surfaces*

14:45 - 14:45 D. van Straten: *The Geometry and Arithmetic of Calabi-Yau operators: *

coffee

16:15 - 17:15 C. Hacon: *Existence of log canonical closures*

### ABSTRACTS OF INVITED TALKS

**Ingrid Bauer**: *Inoue surfaces and Inoue type manifolds: weak rigidity*.

Among minimal surfaces of general type with $p_g = 0$, there is only one class with $K^2=7$, discovered by Inoue in the early nineties. In recent work with Fabrizio Catanese, we show that these surfaces form an irreducible connected component of the moduli space, and that weak rigidity holds for them. Weak rigidity for $X$ means here, that every other variety $Y$ homotopically equivalent to $X$ has the property that either $Y$

or the conjugate variety belongs to an irreducible family containing $X$. We show this result by giving a different description of Inoue surfaces. This description lends itself to generalizations, which I will discuss during the talk. We define a classical Inoue type manifold as an ample divisor in a product of manifolds in the following list: Abelian varieties and quotients of irreducible locally symmetric spaces (including curves). Under

some further assumptions we prove a similar theorem to the above, i.e., that manifolds homotopically equivalent to Inoue-type manifolds are again Inoue-type manifolds.

**Francis Brown:** *On the motivic fundamental group of ${\Bbb P}^1$ minus $3$ points*

I will talk about some recent results and open problems concerning the motivic fundamental group of the projective line minus three points, and the mysterious role played by period polynomials of modular forms.

**Alessio Corti**: *Algebraic geometry and $G_2$ manifolds*

I explain how to construct many examples of compact $7$-dimensional manifolds $M$ with holonomy the exceptional Lie group $G_2$, and associative submanifolds in them, starting from a pair of "asymptically cyclindrical" Calabi--Yau complex $3$-folds $Y_1$ and $Y_2$, and glueing $Y_1*S^1$ to $Y_2*S^1$. In some cases it is possible to determine the diffeomorphism type of $M$ and construct $G_2$ manifolds that are homeomorphic but not diffeomorphic. (Work with M Haskins, J Nordström and T Pacini.)

**Christopher D. Hacon:** *Existence of log canonical closures*

Log canonical singularities are a natural class of singularities arising in higher dimensional algebraic geometry (as well as several other contexts). In this talk we will describe recent joint work with Chenyang Xu on the existence of log canonical compactifications for open $\log$ canonical pairs and applications to the properness of the moduli space of varieties of ($\log$)general type and the existence of $\log$ canonical flips.

**Klaus Hulek**: *On the cohomology of toroidal compactifications of ${\cal A}_g$*

In this talk we will discuss some old and new results on the cohomology of ${\cal A}_g$ and its toroidal compactifications. There are two aspects to this question. The first is to compute the full cohomology in low genus. Here we shall discuss recent results in genus $4$ (joint with O. Tommasi). The second is the question whether one can, as in the case of ${\cal A}_g$ itself, expect stabilization of the cohomology. Here we shall report on work in progress with S. Grushevsky and O. Tommasi.

**Adrian Langer:** *Vanishing theorems for nef line bundles*

I will survey results and counterexamples concerning vanishing theorems for cohomology groups of line bundles on varieties defined over fields of positive characteristic. I will also study a new conjecture concerning

cohomology groups of line bundles on complex varieties and relate it to the Grothendieck-Katz conjecture.

**Sebastian Petersen**: * Independence of $\ell$-adic Galois representations over function fields.*

Let $K$ be a field. For every rational prime $\ell$ let $\rho_\ell: G_K\to \Gamma_\ell$ be a homomorphism into a group $\Gamma_\ell$ and consider the homomorphism $\rho: G_K\to \prod_\ell \Gamma_\ell$ induced by the $\rho_\ell$. We call the family $(\rho_\ell)_\ell$ of homomorphisms {\sl almost independent} if there exists a finite separable extension $E/K$ such that $\rho(G_E)=\prod\limits_\ell \rho_\ell(G_E)$.

Important examples of such families are given by the representations $\rho_{X, \ell}^{(q)}: G_K\to \text{Aut}(H^q(X_{\overline{K}}, {\Bbb Q}_\ell))$ of $G_K$ on the $\ell$-adic \'etale cohomology of a separated algebraic scheme $X/K$. By a recent theorem of Serre the family $(\rho_{X, \ell}^{(q)})_\ell$ is almost independent for every separated algebraic scheme $X$ over a {\sl number field} $K$. The special case where $X$ is an abelian variety over a number field was solved by Serre already in the 80's. Serre and Illusie asked whether these results can be extended to the case of a ground field $K$ which is a finitely generated extension of ${\Bbb Q}$ of transcendence degree $\text{trdeg}(K/{\Bbb Q})\ge 1$. In brief: We answer this question affirmatively. Furthermore we have results in the case of a finitely generated ground field $K$ of positive characteristic. In positive characteristic certain adaptions are necessary already in the statements. (Joint work with Gebhard Böckle and Wojciech Gajda).

**Jürgen Ritter**: *Equivariant Iwasawa Theory*

The talk will first explain the need for proving a 'main conjecture' in non-commutative Iwasawa theory (i.e., Iwasawa theory for Galois extensions $K/k$ of number fields without assuming that the Galois group is abelian) and then somehow sketch its proof by showing how it is based on Wiles' proof of the Main Conjecture in the classical case, on the Deligne-Ribet q-expansion principle for Hilbert modular forms, and on the work by Fr{\" o}hlich and Taylor on the Galois structure of rings of integers in tame extensions $K/k$. The audience will not be bothered by going into technical details; however, those items and ideas which have not appeared in the work just mentioned will have to be explained.

**Matthias Schütt**: *Picard numbers of quintic surfaces*}

The Picard number is a non-trivial invariant of an algebraic surface which captures much of its inner structure. We will discuss the fundamental problem which Picard numbers occur on surfaces on general type for the prototype example of quintics in ${\Bbb P}^3$. We will review what seems to be known and introduce a new technique based on arithmetic deformations which allows us to engineer quintics with prescribed Picard number.

**Bernd Siebert**: *Generalized Theta Functions*

Traditional theta functions provide a canonical basis of sections of the polarizing line bundle on an abelian variety. Through my canonical degeneration approach with Mark Gross on mirror symmetry such functions can now be defined in much greater generality on many varieties with effective anticanonical class such as certain Calabi-Yau and Fano varieties. While the precise nature of these functions is not understood yet, there are many pointers that they play a pivotal role in many applications, including more classical questions in algebraic geometry (joint work with Mark Gross, Paul Hacking, Sean Keel).

**Duco van Straten**: *The Geometry and Arithmetic of Calabi-Yau operators*

The periods of one-parameter families of Calabi-Yau varieties give rise to a Variation of Hodge Structures over the $\mathbb Z$ with Hodge numbers $(1,1,1,1)$ and can be described with the help of its Picard-Fuchs operator. We give examples of constructions and discuss a recent example of such a variation without a point of maximal unipotent monodromy.

**Emmanuel Ullmo**: *The hyperbolic Ax-Lindemann conjecture for projectve Shimura varieties and applications to the André-Oort conjecture.*

We will explain the statement of the hyperbolic Ax-Lindemann conjecture for Shimura varieties. The main result is a proof of this conjecture for projective Shimura varieties (joint workwith Andrei Yafaev). We will also give some applications to the André-Oort conjecture.We'll explain a proof of the André-Oort conjecture for projective Shimura varieties contained in an arbitrary power of the moduli space of principally polarized abelian varieties of dimension $6$.

### ABSTRACTS OF CONTRIBUTED TALKS

**Sara Arias-de-Reyna**: *Compatible systems of symplectic Galois representations with large residual image*

This talk addresses the problem of finding conditions on compatible systems of symplectic Galois representations that ensure that the residual image of all but finitely many of the members of the family is huge (i.e., contains a symplectic group of full dimension). To study the residual image of a symplectic Galois representations, the first step is to analyze the subgroups of $GSp(n, k)$, where $k$ is a finite field. We will see that the subgroups containing a transvection can be classified in three families, one of which consists of the huge subgroups. If we consider Galois representations with an appropriated maximally induced place and satisfying an extra local condition, the classification yields that the residual image is a huge subgroup. The existence of compatible systems of symplectic Galois representations of $G_Q$ satisfying the conditions we describe has applications to the inverse Galois problem over $Q$. Namely, given natural numbers $n$ and $d$ with $n$ even, it would provide us with realizations of $PGSp(n, l^d)$ or $PGSp(n, l^d)$ as Galois groups over $Q$ for a positive density set of primes $l$.

**Andrei Halanay: ***Moduli spaces of vector bundles on principal elliptic fibrations*

We shall present some results on the moduli spaces of relatively semi-stable vector bundles over a principal elliptic fibration having trivial canonical bundle. The main technical tool used in the description of the moduli space will be a twisted Fourier-Mukai transform, an adaptation to our case of that constructed by A. C\u ald\u araru in his thesis.

**Nivaldo Medeiros**: *On the polar degree of projective hypersurfaces*

Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.

**Bartosz Naskręcki**: *Mordell-Weil ranks of families of elliptic curves associated to **Pythagorean triples*

We study a new family of elliptic surfaces related to Pythagorean triples and Frey curves which are neither rational, nor K3. Our goal is to present the method of computing the exact Mordell-Weil rank of the generic fiber which uses \'etale cohomology and good reduction to positive characteristic. We combine numerical computation of the Frobenius characteristic polynomial with Weil conjectures to estimate the upper bound of the rank of N\'eron-Severi group for several intermediate elliptic surfaces.

The conference is partially supported by the City of Cracow within the program Cracow Scientific Conferences. |