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Flows on Shimura varieties

Giacomini, Michele; (2021) Flows on Shimura varieties. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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The aim of this thesis is to study both holomorphic and algebraic flows on Shimura varieties. The first part of the thesis studies holomorphic flows, the main result is a hyperbolic analogue of the Bloch-Ochiai Theorem in the context of mixed Shimura varieties. This extends previous results of Ullmo and Yafaev for co-compact pure Shimura varieties. The proof follows the template set by the hyperbolic Ax-Linedmann-Weierstrass theorem of using the PilaWilkie counting theorem together with some volume inequalities to prove our result. The heart of the proof consists of two volume inequalities, first one for the intersection of a definable set with hyperbolic balls in Hermitian symmetric domains of non-compact type. The second for the intersection of a definable portion of a holomorphic curve with a fundamental domain for the action of a congruence group on a Hermitian symmetric domain of non-compact type. In the second part we study totally geodesic subvarieties of mixed Shimura varieties and algebraic flows. We show that contrary to the case of pure Shimura varieties, there is in general no inclusion either way between the concept of weakly special and totally geodesic subvariety in the mixed setting. Then we report an argument communicated by N. Mok which shows that unlike in the pure case there are totally geodesic submanifolds of a mixed Shimura variety that are not homogeneous. Finally we use these results on totally geodesic subvarieties to state and prove a generalisation of results of Ullmo and Yafaev on algebraic flows on pure Shimura varieties to the mixed case. The proof follows the pure case and uses a theorem of Ratner in arithmetic dynamics.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Flows on Shimura varieties
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10127040
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