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Gromov compactness in non-archimedean analytic geometry

Yu, Tony Yue (2018) Gromov compactness in non-archimedean analytic geometry. Journal Für Die Reine und Angewandte Mathematik, 2018 (741). pp. 179-210. ISSN 0075-4102. doi:10.1515/crelle-2015-0077.

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Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

Item Type:Article
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URLURL TypeDescription Paper
Yu, Tony Yue0000-0002-6019-8552
Alternate Title:Gromov compactness in tropical geometry and in non-Archimedean analytic geometry
Additional Information:© Walter de Gruyter GmbH 2016. Published by De Gruyter January 14, 2016. I am very grateful to Maxim Kontsevich for inspirations and guidance. Special thanks to Antoine Chambert-Loir who provided me much advice and support. I appreciate valuable discussions with Ahmed Abbes, Denis Auroux, Pierrick Bousseau, Olivier Debarre, Antoine Ducros, Lie Fu, Ilia Itenberg, François Loeser, Johannes Nicaise, Mauro Porta, Matthieu Romagny, Michael Temkin and Jean-Yves Welschinger. I would like to thank the referees for helpful comments.
Issue or Number:741
Classification Code:2010 Mathematics Subject Classification: Primary 14G22; Secondary 14D23 14D15 14H10 14T05
Record Number:CaltechAUTHORS:20210914-164413051
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110834
Deposited By: George Porter
Deposited On:14 Sep 2021 22:10
Last Modified:14 Sep 2021 22:11

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