Tame topology of arithmetic quotients and algebraicity of Hodge loci

Bakker, B.; Klingler, B.; Tsimerman, J.

doi:10.1090/jams/952

Tame topology of arithmetic quotients and algebraicity of Hodge loci
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J. Amer. Math. Soc. 33 (2020), 917-939 Request permission

Abstract:

In this paper we prove the following results:

$1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures.

$2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb {V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure.

$3)$ As a corollary of $2)$ and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of $(S, \mathbb {V})$ is a countable union of algebraic subvarieties of $S$, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid.

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