Abstract
We prove a continuity result for the fibers of the Berkovich analytification of a complex algebraic variety with respect to the maximum of the Archimedean norm and the trivial norm. As a consequence, we obtain generalizations of a result of Mikhalkin and Rullgård about degenerations of amoebae onto tropical varieties.
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Notes
We use negative signs to match the standard convention for valuations. All logarithms are natural logarithms.
The superscript “\(h\)” stands for “holomorphic”.
While we shall only consider the analytification as a topological space, one can also equip it with a structure sheaf.
They are good \(k\)-analytic spaces without boundary.
As with the case of the analytification, the tropicalization \({Y^\mathrm {trop}}\) will only be considered as a topological space (together with an action by \({\mathbf {R}}_+^*\)) and not equipped with a structure sheaf.
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Acknowledgments
I thank V. Berkovich for the proof of Lemma 3.2, and M. Baker, S. Boucksom, A. Ducros, W. Gubler, S. Payne and A. Werner, for comments on a preliminary version of this manuscript. I have also benefitted from discussions with E. Brugallé, C. Favre, G. Mikhalkin and B. Teissier.
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Partially supported by NSF Grants DMS-1001740 and DMS-1266207.
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Jonsson, M. Degenerations of amoebae and Berkovich spaces. Math. Ann. 364, 293–311 (2016). https://doi.org/10.1007/s00208-015-1210-3
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DOI: https://doi.org/10.1007/s00208-015-1210-3