On the geometric André–Oort conjecture for variations of Hodge structures

Jiaming Chen

doi:10.1515/crelle-2021-0011

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On the geometric André–Oort conjecture for variations of Hodge structures

Jiaming Chen

Published/Copyright: April 2, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 776

Abstract

Let 𝕍 be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for (S,𝕍), which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

Acknowledgements

I would like to express my deep gratitude to my supervisor Bruno Klingler, who introduced me to the study of the distribution of Hodge locus and suggested that it might be possible to apply the equidistribution methods to it. I very much thank him for his meticulous proofreading and corrections on an early draft of this paper. I want to thank Patrick Brosnan, Carlos Simpson and Emmanuel Ullmo, for their very careful reading of this manuscript and the suggestions for improvements. This work is part of the author’s PhD thesis at the Université de Paris, and part of it were written while visiting Humboldt-Universität zu Berlin. The author wants to thank these institutions for their support and hospitality.

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Received: 2020-12-01

Published Online: 2021-04-02

Published in Print: 2021-07-01

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https://doi.org/10.1515/crelle-2021-0011