Gromov compactness in non-archimedean analytic geometry
- Creators
- Yu, Tony Yue
Abstract
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.
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.Attached Files
Accepted Version - 1401.6452.pdf
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Additional details
- Alternative title
- Gromov compactness in tropical geometry and in non-Archimedean analytic geometry
- Eprint ID
- 110834
- Resolver ID
- CaltechAUTHORS:20210914-164413051
- Created
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2021-09-14Created from EPrint's datestamp field
- Updated
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2021-09-14Created from EPrint's last_modified field