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Tropicalization of the moduli space of stable maps

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Abstract

Let X be an algebraic variety and let S be a tropical variety associated to X. We study the tropicalization map from the moduli space of stable maps into X to the moduli space of tropical curves in S. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in X, the associated tropical curves in S deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.

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Notes

  1. If the residue field \({\tilde{k}}\) has characteristic zero and if X is compact quasi-smooth and strictly \(k\)-analytic, then strictly semi-stable formal models exist up to passing to a finite extension of k (cf. [44, 65]).

  2. We use the terminology in [67], which is different from [9, 10].

  3. It is based on the work of Kontsevich and Tschinkel [47], see also [16].

  4. We refer to [1, 14, 15, 28] for the notion of flatness in non-archimedean analytic geometry.

  5. It is a finite constant group when the field k has characteristic zero.

  6. To evaluate the norm of a D-function on a point of the k-analytic space \(\mathbb D^m\times (\mathbb D^\circ )^n\), it suffices to pass to a ground field extension making the point rational.

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Acknowledgments

I am very grateful to Maxim Kontsevich and Antoine Chambert-Loir for inspirations and support. Special thanks to Antoine Ducros from whom I learned model theory and its applications to tropical geometry. I appreciate valuable discussions with Vladimir Berkovich, Pierrick Bousseau, Ilia Itenberg, François Loeser, Florent Martin, Johannes Nicaise, Sam Payne and Michael Temkin. Comments given by the referees helped greatly improve the paper.

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  1. Institut de Mathématiques de Jussieu - Paris Rive Gauche, CNRS-UMR 7586, Case 7012, Université Paris Diderot - Paris 7, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Tony Yue Yu

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Yu, T.Y. Tropicalization of the moduli space of stable maps. Math. Z. 281, 1035–1059 (2015). https://doi.org/10.1007/s00209-015-1519-3

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