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Tame topology of arithmetic quotients and algebraicity of Hodge loci


Authors: B. Bakker, B. Klingler and J. Tsimerman
Journal: J. Amer. Math. Soc. 33 (2020), 917-939
MSC (2010): Primary 14D07; Secondary 14C30, 22F30, 03C64
DOI: https://doi.org/10.1090/jams/952
Published electronically: September 15, 2020
MathSciNet review: 4155216
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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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Additional Information

B. Bakker
Affiliation: Department of Mathematics, University of Georgia, 452 Boyd Graduate Studies, Athens, Georgia 30602
MR Author ID: 920702
Email: bakker.uga@gmail.com

B. Klingler
Affiliation: Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6 - 10099 Berlin
MR Author ID: 611580
Email: bruno.klingler@hu-berlin.de

J. Tsimerman
Affiliation: Department of Mathematics, University of Toronto, 215 Huron Street, Toronto, Canada M5S 1A2
MR Author ID: 896479
Email: jacobt@math.toronto.edu

Received by editor(s): October 1, 2018
Received by editor(s) in revised form: September 27, 2019, and September 28, 2019
Published electronically: September 15, 2020
Additional Notes: The first author was partially supported by NSF grants DMS-1702149 and DMS-1848049.
The second author was partially supported by an Einstein Foundation’s professorship.
Article copyright: © Copyright 2020 American Mathematical Society