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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Tame topology of arithmetic quotients and algebraicity of Hodge loci
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by B. Bakker, B. Klingler and J. Tsimerman HTML | PDF
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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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.
  • © Copyright 2020 American Mathematical Society
  • 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
  • MathSciNet review: 4155216