Abstract
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.
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Notes
It is essential to restrict to Griffiths transverse subvarieties, as the general statement is false since, for example, D contains compact subvarieties.
Strictly speaking, a compactification of X should be chosen.
Recall that very general means in the complement of countably many proper closed subvarieties.
Recall that for a function f(R), saying \(f(R)=\omega (R)\) means that for some positive constant \(\delta >0\) we have \(f(R)\ge \delta R\).
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Acknowledgements
The first author was partially supported by NSF grant DMS-1702149. We would like to thank the referee for helpful comments and for suggestions which improved the readability of the paper.
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Bakker, B., Tsimerman, J. The Ax–Schanuel conjecture for variations of Hodge structures. Invent. math. 217, 77–94 (2019). https://doi.org/10.1007/s00222-019-00863-8
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DOI: https://doi.org/10.1007/s00222-019-00863-8