Belgian-Dutch Algebraic Geometry Seminar

Program:

• 13:30 - 14:30: Gregorio Baldi (Paris).  Special Points in Families of Algebraic Varieties

• Coffee Break;

• 15:00-16:00:  Joana Cirici (Barcelona)  Configuration spaces of algebraic varieties

• 16:30-17:30: Salvatore Floccari (Berlin)  On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type

• 17:30-19:00: Reception: Registration's link before 1st December

Abstracts below.

Abstracts:

Gregorio Baldi: Special Points in Families of Algebraic Varieties

Given a family of algebraic varieties defined over a number field, a natural question is how the points whose fibres have special geometric or arithmetic properties are distributed. This talk will survey recent results on the distribution of the Hodge locus and explore their connections with the Hodge, Mumford-Tate and Grothendieck period conjectures.

Joana Cirici: Configuration spaces of algebraic varieties

A long-standing question in algebraic topology asks whether the homotopy type of the configuration space Conf_n(X) of n ordered points in a closed, simply connected manifold X depends only on the homotopy type of X. In this talk I will explain how Galois actions on étale cohomology can be used to prove strong results in this direction when X is a smooth projective complex variety. I will in fact discuss the more general setting of complements of arrangements of subvarieties in a smooth projective variety. This places the Kriz-Totaro models for configuration spaces and Morgan's results on the homotopy types of smooth varieties within a single framework. This is joint work with Geoffroy Horel.

Salvatore Floccari: On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type.

The Hodge conjecture is one of the most famous open problems in complex algebraic geometry. I will present some new results on the conjecture for hyper-Kähler varieties, and in particular for those of generalized Kummer type, which constitute one of the two infinite series of examples of hyper-Kähler varieties currently known. 

I will explain how, in a joint work with Lie Fu, we are able to prove the Hodge conjecture for all varieties of generalized Kummer type of dimension divisibile by 4, and all of their powers. This strong result rests on ingredients established over the years by Hassett-Tschinkel, Foster, O'Grady, Markman, Voisin, Varesco as well as ourselves.