Conférence GAGC 2016 (en 2017)

Liste des contributions

7. Enumeration of curves on K3 surfaces by polyhedral degenerations

Thomas Dedieu

Let $(S,L)$ be a primitively polarized K3 surface, $k$ an integer. Integral curves of geometric genus $g$ in the linear system $|kL|$ form a family of dimension $g$ (if non-empty). One wants to count the number of such curves passing through $g$ general points fixed on $S$. Gromov-Witten theory provides a complete answer to this question when $k=1$, but poses serious problems when$...

13. Finiteness results for real structures on rational surfaces

Mohamed Benzerga

A real structure on a complex projective variety X is an antiregular (or antiholomorphic) involution. The data of such a structure on X is equivalent to the data of a real variety whose complexification is isomorphic to X (i.e. a real form of X). The aim of this talk is to show how the study of automorphism groups of rational surfaces can be used in order to give a partial answer to the...

14. Liouville's inequality for transcendental points on projective varieties

Carlo Gasbarri

Liouville inequality is a lower bound of the norm of an integral section of a line bundle on an algebraic point of a variety. It is an important tool in may proofs in diophantine geometry and in transcendence. On transcendental points an inequality as good as Liouville inequality cannot hold. We will describe similar inequalities which hold for "many" transcendental points and some applications

11. Maximal representations of uniform complex hyperbolic lattices

Julien Maubon

Let $\Gamma$ be a uniform complex hyperbolic lattice, that is, a discrete subgroup of the group of biholomorphisms $PU(n,1)$ of the ball $B^n$ acting cocompactly on $B^n$. If $\rho$ is a representation (a group homomorphism) of $\Gamma$ in a semisimple Lie group of Hermitian type $G$, the Toledo invariant of $\rho$ is a measure of the « complex size » of $\rho$. It is bounded by a quantity...

5. Mini-cours : Birational geometry of moduli spaces

Radu Laza (Stony Brook University)

Overview: The ideal situation in moduli theory is to have a geometric/modular compactification for a given moduli space, and to have a good understanding of its structure so that one can compute various invariants (e.g. Betti numbers). Unfortunately, this is rarely the case (e.g. for varieties of general type, there is a geometric compactification, the so called KSBA compactification, but we...

15. Mini-cours : Dynamical degrees of birational transformations of surfaces

Jérémy Blanc

The dynamical degree $\lambda(f)$ of a birational transformation $f$ of a surface measures the exponential growth of the formula that define the iterates of $f$. It allows to study the complexity of the dynamic of $f$. For instance, over the field of complex number, the number $\log(\lambda(f))$ gives a upper bound for the topological entropy. The number $\lambda(f)$ is an algebraic integer of...

16. Mini-cours : Log Calabi-Yau varieties, degenerations of Calabi-Yau varieties, and mirror symmetry

Mark Gross

I will try to provide some context and background for recent breakthroughs by Siebert and myself involving constructions of mirrors to various types of varieties. In particular, I will give a gentle introduction to logarithmic Gromov-Witten theory and explain how log GW invariants can be used to generalize constructions of Gross-Siebert and Gross-Hacking-Keel to give a very general...

18. Mini-cours : Rationalité stable

Alena Pirutka

Soit X une variété algébrique complexe, projective et lisse. On dispose de plusieurs notions pour déterminer si X est 'proche' à un espace projectif : la variété X est rationnelle si un ouvert de X est isomorphe à un ouvert d'un espace projectif, on dit que X est stablement rationnelle si cette propriété vaut en remplaçant X par un produit avec un espace projectif, enfin X est unirationnelle...

17. Mini-cours : The non-archimedean SYZ fibration

Johannes Nicaise

The SYZ conjecture gives a geometric description of the relation between mirror pairs of Calabi-Yau varieties. It was a fundamental insight of Kontsevich and Soibelman that the structures predicted by the SYZ conjecture can be found in the world of non-archimedean geometry (Berkovich spaces). I will explain some of the main ideas, as well as the connections with the minimal model program in...

9. On the unirationality of Hurwitz spaces

Fabio Tanturri

In this talk I will discuss about the unirationality of the Hurwitz spaces $H_{g,d}$ parametrizing d-sheeted branched simple covers of the projective line by smooth curves of genus $g$. I will summarize what is already known and formulate some questions and speculations on the general behaviour. I will then present a proof of the unirationality of $H_{12,8}$ and $H_{13,7}$, obtained via...

12. Perverse motives and mixed Hodge modules

Florian Ivorra

Let X be a smooth complex algebraic variety. In this talk, I will explain a way to use perverse homology sheaves of families of algebraic varieties over X to extend Nori’s construction of an Abelian category of motives to a relative setting. This approach (which may also be applied to perverse sheaves over finite field) leads to a notion of motivic perverse sheaves and provides a mean to...

10. Sur l'amplitude du cotangent des intersections complètes

Lionel Darondeau

C'est un travail commun avec Damian Brotbek. Nous prouvons que toute variété projective lisse $M$ contient des sous-variétés avec cotangent ample en toute dimension $n\leq dim(M)/2$. Nous construisons de telles variétés comme certaines intersections complètes.

8. The eventual paracanonical map of an irregular variety.

Rita Pardini

I will report on joint work with. M.A. Barja (UPC, Barcelona) and L. Stoppino (Universita' dell'Insubria, Italy) Given a map $a:X\longrightarrow A$ from a smooth projective variety to an abelian variety and a line bundle $L$ on $X$, we study the "eventual" behaviour of the linear system $|L|$ under base change with the $d$-th multiplication map $A\longrightarrow A$. We prove a...

6. Twisted Kodaira-Spencer classes and their use in the study of invariants of surfaces in $\mathbb{P}^4$

Daniel Naie

(joint work Igor Reider) Let $X$ be a projective surface. A twisted Kodaira-Spencer class is an element of the cohomology group $H^1(T_X(-D))$, with $D$ ``sufficiently positive''. We study the connection between the existence of a non-trivial twisted class and the geometry of $X$. In particular, we show that, for a minimal general type surface satisfying $c_2/c_1^2<5/6$, the...