# Activités organisées par SÉDIGA

## Mai 2011

• Colloque à Nice
Date et durée: 25-27 mai 2011
Thème: Topologie des variétés algébriques
Organisateur: Alexandru Dimca
Contact: Alexandru Dimca
Inscription préalable nécessaire avant le 25 avril 2011: le site d'inscription est ouvert. Il est nécessaire de se pré-inscrire sur ce site, puis de confirmer l'inscription après la réception d'un courrier électronique d'acceptation.

• Conférenciers invités:
D. Arapura, I. Bauer, A. Beauville, F. Campana, F. Catanese, P. Py, B. Klingler, Mahan Mj, S. Papadima, A. Suciu, C. Voisin

Programme:

Mercredi 25 mai
9h30-10h  Accueil
10h-11h  Arnaud Beauville: The Lüroth problem and the Cremona group
Résumé: The Lüroth problem asks whether every field $$K$$ containing $$\mathbb C$$ and contained in $$\mathbb C(x_1,\dots,x_n)$$ is of the form $$\mathbb C(y_1,\dots,y_p)$$. After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example. Finally I will explain the relation with the study of the finite groups of birational automorphisms of $$\mathbb P^3$$.
11h-11h30 Pause café
11h30-12h30  Pierre Py: Kähler groups, real hyperbolic spaces and the Cremona group
Résumé: Starting from a classical theorem of Carlson and Toledo, we will discuss actions of fundamental groups of compact Kähler manifolds on finite or infinite dimensional real hyperbolic spaces. We will see that such actions almost always (but not always) come from surface groups. We then give an application to the study of the Cremona group. This is a joint work with Thomas Delzant.
14h30-15h30 Fabrizio Catanese: Special Galois coverings and the irreducibility of certain spaces of coverings of curves, with applications to the moduli space of curves
Résumé: Special Galois coverings are e.g. cyclic or dihedral coverings, for which I will describe old and new results, and new examples, obtained together with Fabio Perroni and Michael Loenne. In the case of curves I will show some irreducibility results for coverings of a fixed numerical type: in the cyclic case for smooth curves, and in the cyclic case of prime order for moduli-stable curves. In the dihedral case we have results in work in progress with Michael Loenne and Fabio Perroni: in the case where the genus of the base is 0, or in the case where the covering is étale. In this case our work ties in with some general asymptotical study done by Dunfield and Thurston. One application of the cyclic case is the description of an irredundant irreducible decomposition for the singular locus of the compactified Moduli space of curves of genus $$g$$, extending the result of Cornalba for the open set $$M_g$$.
15h30-16h Pause café
16h-17h Alexandru Suciu: Abelian Galois covers and rank one local systems
Résumé: The Galois covers of a connected, finite CW-complex $$X$$ with group of deck transformations a fixed Abelian group admit a natural parameter space, which in the case of free abelian covers of rank $$r$$ is simply the Grassmannian of $$r$$-planes in $$H^1(X,\mathbb Q)$$. The Betti numbers of such covers are determined by the jump loci for homology with coefficients in rank $$1$$ local systems on $$X$$, and the way these loci intersect with certain algebraic subgroups in the character group of $$\pi_1(X)$$. Under favorable circumstances, the finiteness of those Betti numbers is controlled by the jump loci of the cohomology ring of $$X$$. In this talk, I will discuss this circle of ideas, and give some new examples where such computations play a role, especially in the case when $$X$$ is a smooth, quasi-projective complex variety.

Jeudi 26 mai
10h-11h Bruno Klingler: Symmetric differentials and Kähler groups
Résumé: I will discuss the relation between rigidity properties for the fundamental group of a smooth projective variety $$X$$ and the structure of symmetric holomorphic differentials on $$X$$.
11h-11h30 Pause café
11h30-12h30  Alexandru Dimca:  Milnor fibres of hyperplane arrangements
Résumé:
14h30-15h30 Stefan Papadima: Diophantine geometry, representation theory and homology of the Johnson filtration
Résumé: I will present answers to questions raised by B. Farb and F. Cohen, concerning the homology of the second Johnson subgroup of Torelli groups. The approach is based on the representation theory of arithmetic groups, on affine tori and their Lie algebras. This is joint work with A. Dimca, R. Hain and A. Suciu.
15h30-16h Pause café
16h-17h  Mahan MJ: Three manifolds groups, Kähler groups and complex surfaces
Résumé: Let $$1 \to N \to G \to Q \to 1$$ be an exact sequence of finitely presented groups, where $$Q$$ is infinite and not virtually cyclic, and is the fundamental group of some closed 3-manifold.  If $$G$$ is Kähler, we show that $$Q$$ contains as a finite index subgroup either a finite  index subgroup of the 3-dimensional Heisenberg group, or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. If $$G$$ is the fundamental group of a compact complex surface, we show that $$Q$$ must contain the fundamental group of a Seifert-fibered three manifold as a finite index sub-group, and $$G$$ contains as a finite index subgroup the fundamental group of an elliptic fibration. This is joint work with I. Biswas and H. Seshadri.

Vendredi 27 mai
10h-11h  Claire Voisin: The decomposition theorem for families of K3 surfaces and Calabi-Yau hypersurfaces
Résumé: The decomposition theorem for smooth projective morphisms $$\pi: X\to B$$ says that $$R\pi_*\mathbb Q$$ decomposes as $$\oplus_i R^i\pi_*\mathbb Q[-i]$$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $$B$$. We prove however that this is always possible for families of K3 surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of K3 surfaces. We also prove that such a multiplicative decomposition isomorphism exists for Calabi-Yau hypersurfaces in $$\mathbb P^n$$.
11h-11h30 Pause café
11h30-12h30  Frédéric Campana: Conjecture d'abélianité pour les variétés kählériennes compactes "spéciales"
Résumé: Les variétés kählériennes (compactes) spéciales sont celles ne dominant pas d'"orbifoldes" de type général. Elles généralisent les courbes rationnelles et elliptiques en toutes dimensions, et sont antithétiques des variétés de type général. Toute variété kählériennes compacte $$X$$ se décompose canoniquement et fonctoriellement à l'aide d'une fibration (son "cœur") en ses parties "spéciales" (les fibres), et de type général (la "base orbifolde"). Cette décomposition fournit conjecturalement un "scindage" des propriétés de $$X$$ (aux niveaux hyperbolique, arithmétique si $$X$$ est projective, et topologique). Par exemple, on conjecture que le groupe fondamental de $$X$$ est virtuellement abélien si $$X$$ est "spéciale". Cette conjecture est vraie si ce groupe fondamental est soit linéaire, soit résoluble. Nous démontrons (travail en commun avec B. Claudon) que c'est le cas si $$X$$ est de dimension au plus $$3$$ en utilisant des arguments métriques (Calabi-Yau orbifolde) et le programme des modèles minimaux en dimension $$2$$.
14h30-15h30 Ingrid Bauer: Rational curves on product-quotient surfaces
Résumé:
15h30-16h Pause café
16h-17h Donu Arapura: Nori's Hodge conjecture
Résumé: Nori's conjecture, which is not so well known, says that his category of motives embeds fully and faithfully into the category of mixed Hodge structures. This should be viewed as a refinement of Deligne's absoluteness conjecture. I want to explain the conjecture, and then explain how to prove the special case for the tensor subcategory generated by smooth affine curves, which contains things like semi-abelian varieties.