Hodge theory and o-minimality at CIRM
Gregorio Baldi
TL;DR
This work surveys the interplay between o-minimality, Ax-Schanuel type functional transcendence, and the Zilber-Pink philosophy in the setting of variations of Hodge structures (VHS). It develops a cohesive framework linking period maps to finiteness and algebraicity results: CDK’s Hodge locus structure, atypical/typical intersections, and the geometric Zilber-Pink conjecture for VHS, all underpinned by Ax-Schanuel and definability in o-minimal structures. Key contributions include a proof outline for finiteness of atypical loci via AS plus degenerations, definable fundamental sets in Hodge theory, and stronger AS variants via period torsors and foliated structures, with culminating results on algebraicity and quasi-projectivity of period-map images (via definable GAGA). The practical impact lies in providing a unified, technically robust approach to long-standing questions about when Hodge loci are algebraic, how period maps behave, and how VHS data constrain arithmetic and geometric structures across families of varieties. The synthesis offers a roadmap for applying these ideas to Noether-Lefschetz loci, Jacobian problems, and arithmetic questions about integral points and rational sections in families. The discussion also highlights recent advances by Bakker, Brunebarbe, and Tsimerman, illustrating the power of combining o-minimality with Hodge theory to obtain deep structural results.
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.