Abstract
We prove that compact Berkovich spaces are also sequentially compact when defined over the field of formal Laurent series in one variable. The proof is based on the extensive use of the Riemann-Zariski space of a variety.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Supported by the ERC-project ‘Nonarcomp’ no.307856., by the ANR project Berko, and by the ECOS project C07E01.
- 2.
A separable space is a topological space admitting a countable dense subset. We shall never use the notion of separable field extension so that no confusion should occur.
- 3.
This means X is irreducible reduced of finite type and k is algebraically closed. The last assumption is however inessential in our paper.
- 4.
That is any sequence of points in this set has a cluster point in the ambient space.
References
M. Baker, R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line. Mathematical Surveys and Monographs, vol. 159 (American Mathematical Society, Providence, RI, 2010)
V.G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Math. Surveys Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990)
C. Favre, M. Jonsson, The Valuative Tree. Lecture Notes in Mathematics, vol. 1853 (Springer, Berlin, 2004)
K. Floret, Weakly Compact Sets. Lecture Notes in Mathematics, vol. 801 (Springer, Berlin, 1980)
W. Gubler, Local heights of subvarieties over non-archimedean fields. J. Reine Angew. Math. 498, 61–113 (1998)
M. Jonsson, Dynamics on Berkovich Spaces in Low Dimensions, this volume
N. Maïnetti, Sequential compactness of some analytic spaces. J. Anal. 8, 39–54 (2000)
J. Poineau, Les espaces de Berkovich sont angéliques. Bull. de la Soc. Math. France 141(2), 267-297 (2013)
A. Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscripta Math. 123(4), 381–451 (2007)
M. Vaquié, Valuations, in Resolution of Singularities (Obergurgl, 1997). Progress in Math., vol. 181 (Birkhäuser Verlag, Basel, 2000), pp. 539–590
X. Yuan, S.-W. Zhang, Calabi-Yau theorem and algebraic dynamics. Preprint (2009) www.math.columbia.edu/~szhang/papers/Preprints.htm
O. Zariski, P. Samuel, Commutative Algebra, vol. 2. Graduate Texts in Mathematics, No. 29 (Springer, New York-Heidelberg-Berlin, 1975)
Acknowledgements
We thank T. De Pauw, A. Ducros, M. Jonsson, J. Kiwi, J. Nicaise, and R. Menares for useful discussions on the material presented in this paper. Also we deeply thank J. Poineau for kindly informing the author about his proof of the sequential compactness of compact analytic spaces over an arbitrary field.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Favre, C. (2015). Countability Properties of Some Berkovich Spaces. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-11029-5_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11028-8
Online ISBN: 978-3-319-11029-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)