Date and duration: May 25-27, 2011

Topic: Topology of algebraic varieties

Organizer: Alexandru Dimca

Contact:

Registration is necessary before April 25, 2011: the registration form is available. Pre-registration should be done by completing the registration form, and should be confirmed after receiving the acceptation e-mail.

Invited speakers:

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

Program:

Wednesday, May 25

9h30-10h Welcome

10h-11h Arnaud
Beauville:
The Lüroth problem and the Cremona group

Abstract:
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 Coffee break

11h30-12h30 Pierre
Py:
Kähler groups, real hyperbolic spaces and the Cremona group

Abstract:
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

Abstract:
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 Coffee break

16h-17h Alexandru
Suciu: Abelian Galois covers and rank one local systems

Abstract:
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.

Thursday May 26

10h-11h Bruno
Klingler:
Symmetric differentials and Kähler groups

Abstract:
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 Coffee break

11h30-12h30 Alexandru Dimca:
Milnor fibres of hyperplane arrangements

Abstract:

14h30-15h30 Stefan
Papadima:
Diophantine geometry, representation theory and homology of the Johnson
filtration

Abstract:
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 Coffee break

16h-17h Mahan
MJ:
Three manifolds groups, Kähler groups and complex surfaces

Abstract:
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.

Friday May 27

10h-11h Claire
Voisin:
The decomposition theorem for families of K3 surfaces and Calabi-Yau
hypersurfaces

Abstract:
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 Coffee break

11h30-12h30 Frédéric
Campana: Abelianity conjecture for "special" compact Kähler manifolds

Abstract:
The "special" (compact) Kähler manifolds are those which do not dominate an "orbifold" of general type. They generalize the rational elliptic curves in any dimension, and are antithetic to manifolds of general type. Each compact Kähler manifold \(X\) decomposes in a canonical and functorial way through a fibration (its "heart") in its "special" parts (the fibres) and its part of general type (the "orbifold basis"). This decomposition conjecturally furnishes a "splitting" of the properties of \(X\) (at the hyperbolic, arithmetic -if X is projective- and topological levels).
For instance, one conjectures that the fundamental group of \(X\) is virtually abelian if \(X\) is "special". This conjecture is true if the fundamental group is either linear or solvable. We prove (joint work with B. Claudon) that such is the case if X has dimension at most \(3\), by using metrical arguments (Calabi-Yau orbifold) and the minimal model program in dimension \(2\).

14h30-15h30 Ingrid
Bauer: Rational curves on product-quotient surfaces

Abstract:

15h30-16h Coffee break

16h-17h Donu
Arapura:
Nori's Hodge conjecture

Abstract:
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.