The Ax-Schanuel conjecture for variations of Hodge structures

TL;DR

This paper generalizes the Ax-Schanuel theorem to all varieties with pure polarized integral variations of Hodge structures, introducing a volume bound on Griffiths transverse subvarieties as a key new element.

Contribution

It extends the Ax-Schanuel theorem from Shimura varieties to a broader class of Hodge structure varieties, incorporating a novel volume bound technique.

Findings

Proves the Ax-Schanuel conjecture for these varieties.

Establishes a volume bound on Griffiths transverse subvarieties.

Generalizes previous results to new geometric contexts.

Abstract

We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths transverse subvarieties of period domains.