The Ax–Schanuel conjecture for variations of Hodge structures.

We extend the Ax–Schanuel theorem recently proven for Shimura varieties by Mok–Pila–Tsimerman to all varieties supporting a pure polarizable integral variation of Hodge structures. In fact, Hodge theory provides a number of conceptual simplifications to the argument. The essential new ingredient is a volume bound for Griffiths transverse subvarieties of period domains. [ABSTRACT FROM AUTHOR]