The Ax–Schanuel conjecture for variations of Hodge structures

By Benjamin Bakker, Jacob Tsimerman. Inventiones Mathematicae, 2017

70 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f2118fdc3284fd7ba1eb8f333370313f54ec1a4

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Citations in the corpus

Ax-Schanuel for Shimura varieties (2017, unknown): “A generalization of Theorem 1.1 to variations of Hodge structures is given by Bakker-Tsimerman [4].”
Tame topology of arithmetic quotients and algebraicity of Hodge loci (2018, methodology): “In view of Theorem 1.3, its corollary Theorem 1.6, and the recent proof [BaT17] (using Theorem 1.6 and o-minimal technics) of the Ax-Schanuel conjecture for pure Hodge varieties stated in [K17, Conj.”
On the distribution of the Hodge locus (2021, unknown): “Our ﬁrst main result establishes the geometric part of Conjecture 2.5: we prove that the maximal atypical special subvarieties of positive period dimension arise in a ﬁnite number of families whose geometry we control.”