ICTS

This is a followup discussion meeting on complex and algebraic geometry involving the below speakers.

1. Jose Ignacio Butgos Gil
Arakelov theory, equidistribution and algebraic dynamics: The famous Bogomolov conjecture states that, given a curve C of genus bigger or equal than 2 and defined over a number field, then there exists a number ε > 0 such that the set of algebraic points of C whose N eron–Tate height is smaller that ε is finite. This conjecture was proved by Ullmo and Zhang at the end of last century, based on a pioneering work of Szpiro, Ullmo and Zhang on equidistribution of points of small height. Arakelov theory brings geometric intuition to arithmetic and is a natural framework where to study heights and many questions related to equidistribution. In this course we will give an introduction to heights using Arakelov theory, we will discuss classical equidistribution results and their applications, putting on equal foot the arithmetic and the geometric setting. Finally we will explain the recent developments on heights on quasi-projective varieties due to Yuan and Zhang and their application to algebraic dynamics.

2. Bruno Klingler
Hodge thoery, between algebraicity and transcendence: Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained. Tame geometry, which studies structures where every definable set has a finite geometric complexity, appears as a natural setting for understanding these constraints: period maps associated to variations of Hodge structures on complex quasi-projective varieties are tame. I will explain how functional transcendence results for period maps in this setting (Ax-Schanuel for variations of Hodge structures) enable us to prove surprisingly strong algebraicity results for Hodge loci. Time permitting, I will turn to the arithmetic aspects of these questions. Based on the work of Bakker, Baldi, Brunebarbe, Ullmo, Tsimerman and myself.

3. Alessandro Ghigi and Paola Frediani
Differential geometry of the Torelli map: Let j : Mg→Ag be the Torelli map. Outside the hyperelliptic locus j is an orbifold embedding. The goal of the lectures is to relate infinitesimal properties of this embedding at a moduli point [C] with the geometry of the curve C. We will first describe the computation of the second fundamental form using Hodge-Gaussian maps. We will next reinterpret it as a multiplication map on sections and we will derive dimension bounds for totally geodesic subvarieties generically contained in the Jacobian locus. We will also describe the connection with the projective structures on Riemann surfaces. Finally we will consider the Lie bracket on the tangent space of T[C]→g and we will show that its restriction to T[C]M∗g can be interpreted as the multiplication map by the Bergman kernel.

ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.