Hodge theory and o-minimality at CIRM

TL;DR

This paper explores the interplay between o-minimality, Hodge theory, and the Zilber-Pink conjecture, highlighting recent advances in functional transcendence, algebraicity of period maps, and applications to the distribution of Hodge loci.

Contribution

It connects o-minimal geometry with Hodge theory and Diophantine geometry, providing new insights into the distribution of Hodge loci and algebraicity of period map images.

Findings

Recent Ax-Schanuel theorems enhance understanding of functional transcendence.

Distribution of Hodge loci analyzed using o-minimal techniques.

Proven algebraicity and quasiprojectivity of period map images.

Abstract

We discuss the relationship between o-minimality and the so called Zilber-Pink conjecture. Since the work of Pila and Zannier, algebraization theorems in o-minimal geometry had profound impacts in Diophantine geometry (most notably on the study of special points in abelian and Shimura varieties). We will first focus on functional transcendence, discussing various recent and spectacular Ax-Schanuel theorems, and the related geometric part of Zilber-Pink. Armed with these tools, we will study the distribution of the Hodge locus of an arbitrary variation of Hodge structures (the typical/atypical dichotomy) and present some recent applications. We will conclude by describing the algebraicity and quasiprojectivity of images of period maps.