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arXiv · 2005.03524

Special subvarieties of non-arithmetic ball quotients and Hodge Theory

Gregorio Baldi·Emmanuel Ullmo
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Abstract

Let $\Gamma \subset \operatorname{PU}(1,n)$ be a lattice, and $S_\Gamma$ the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove an Ax-Schanuel Conjecture for $S_\Gamma$, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise $S_\Gamma$ inside a period domain for polarised integral variations of Hodge structures and interpret totally geodesic subvarieties as unlikely intersections.

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Gregorio Baldi, Emmanuel Ullmo. 2020-05-07. Special subvarieties of non-arithmetic ball quotients and Hodge Theory. https://arxiv.org/abs/2005.03524

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