Abstract

The goal of this minicourse is to introduce the recent advances in Hodge theory via o-minimal techniques. We will start by revisiting the basics on o-minimal geometry with a view towards algebraization theorems (Pila-Wilkie counting theorem; o-minimal Chow and GAGA theorems in definable complex analytic geometry). In the realm of Hodge theory, we will explore several applications: (i) the definability of period maps (which enables us to easily recover a theorem of Cattani-Deligne-Kaplan on the algebraicity of Hodge loci) and the algebraicity of images of period maps (Griffiths conjecture); (ii) Ax-Schanuel for variations of Hodge structures; (iii) distribution of Hodge loci.