Geodesic
The universal tropicalization and the Berkovich analytification
Kybernetika, Tome 58 (2022) no. 5, pp. 790-815
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element $\mathbb{F}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme $\mathit{Trop}_{univ}(X)$ whose $\mathbb{T}$-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of $X$. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When $X=\mathrm{Spec}\: A$ is affine, we show that $\mathit{Trop}_{univ}(X)$ is the limit of the tropicalizations of $X$ with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $\mathit{Trop}_{univ}(X)$ represents the moduli functor of semivaluations on $X$, and when $X=\mathrm{Spec}\: A$ is affine there is a universal semivaluation on $A$ taking values in the idempotent semiring of regular functions on the universal tropicalization.
Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element $\mathbb{F}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm{an}}$. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme $\mathit{Trop}_{univ}(X)$ whose $\mathbb{T}$-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of $X$. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When $X=\mathrm{Spec}\: A$ is affine, we show that $\mathit{Trop}_{univ}(X)$ is the limit of the tropicalizations of $X$ with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $\mathit{Trop}_{univ}(X)$ represents the moduli functor of semivaluations on $X$, and when $X=\mathrm{Spec}\: A$ is affine there is a universal semivaluation on $A$ taking values in the idempotent semiring of regular functions on the universal tropicalization.
DOI : 10.14736/kyb-2022-5-0790
Classification : 14G22, 14T05
Keywords: tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation 
@article{10_14736_kyb_2022_5_0790,
     author = {Giansiracusa, Jeffrey and Giansiracusa, Noah},
     title = {The universal tropicalization and the {Berkovich} analytification},
     journal = {Kybernetika},
     pages = {790--815},
     year = {2022},
     volume = {58},
     number = {5},
     doi = {10.14736/kyb-2022-5-0790},
     mrnumber = {4538626},
     zbl = {07655860},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-5-0790/}
}
TY  - JOUR
AU  - Giansiracusa, Jeffrey
AU  - Giansiracusa, Noah
TI  - The universal tropicalization and the Berkovich analytification
JO  - Kybernetika
PY  - 2022
SP  - 790
EP  - 815
VL  - 58
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-5-0790/
DO  - 10.14736/kyb-2022-5-0790
LA  - en
ID  - 10_14736_kyb_2022_5_0790
ER  - 
%0 Journal Article
%A Giansiracusa, Jeffrey
%A Giansiracusa, Noah
%T The universal tropicalization and the Berkovich analytification
%J Kybernetika
%D 2022
%P 790-815
%V 58
%N 5
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-5-0790/
%R 10.14736/kyb-2022-5-0790
%G en
%F 10_14736_kyb_2022_5_0790
Giansiracusa, Jeffrey; Giansiracusa, Noah. The universal tropicalization and the Berkovich analytification. Kybernetika, Tome 58 (2022) no. 5, pp. 790-815. doi: 10.14736/kyb-2022-5-0790

[1] Banerjee, S.: Tropical geometry over higher dimensional local fields. J. Reine Angew. Math. 2015 (2013), 71-87. | DOI

[2] Berkovich, V. G.: Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, American Mathematical Society 33, Providence 1990.

[3] Berkovich, V. G.: Smooth $p$-adic analytic spaces are locally contractible. Invent. Math. 137 (1999), 1, 1-84. | DOI

[4] Berkovich, V. G.: Smooth $p$-adic analytic spaces are locally contractible. II. In: Geometric aspects of Dwork theory. (Vol. I, II.), Walter de Gruyter, Berlin 2004, pp. 293-370. | DOI

[5] Deitmar, A.: Schemes over $\mathbb{F}_1$. In: Number fields and function fields-two parallel worlds, Progr. Math. 239, Birkhäuser Boston, Boston, 2005, pp. 87-100.

[6] Deitmar, A.: $\mathbb{F}_1$-schemes and toric varieties. Beiträge Algebra Geom. 49 (2008), 2, 517-525.

[7] Durov, N.: New approach to arakelov geometry. arXiv:0704.2030, 2007.

[8] Einsiedler, M., Kapranov, M., Lind, D.: Non-Archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601 (2006), 139-157.

[9] Foster, T., Gross, P., Payne, S.: Limits of tropicalizations. Israel J. Math. 201 (2014), 2, 835-846. | DOI

[10] Foster, T., Ranganathan, D.: Hahn analytification and connectivity of higher rank tropical varieties. Manuscr. Math. 151 (2016), 3-4, 353-374. | DOI

[11] Giansiracusa, J., Giansiracusa, N.: Equations of tropical varieties. Duke Math. J. 165 (2016), 18, 3379-3433. | DOI

[12] Gubler, W., Rabinoff, J., Werner, A.: Skeletons and tropicalizations. arXiv:1404.7044, 2014.

[13] Hrushovski, E., Loeser, F.: Non-archimedean tame topology and stably dominated types. Ann. Math. Studies 192, Princeton University Press, Princeton 2016.

[14] Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. Aspects of Mathematics E30, Friedr. Vieweg and Sohn, Braunschweig 1996.

[15] Jun, J., Mincheva, K., Tolliver, J.: Vector bundles on tropical schemes. arXiv:2009.03030, 2020.

[16] Kajiwara, T.: Tropical toric geometry. In: Contemp. Math., Toric topology 460, Amer. Math. Soc., Providence 2008, pp. 197-207.

[17] Kato, K.: Toric singularities. Amer. J. Math. 116 (1994), 5, 1073-1099. | DOI

[18] Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: Progr. Math., The Unity Math. 244, Birkhäuser Boston, Boston 2006. pp. 321-385. | DOI

[19] Kontsevich, M., Tschinkel, Y.: Non-archimedean kähler geometry.

[20] Kuronya, A., Souza, P., Ulirsch, M.: Tropicalization of toric prevarieties. arXiv:2107.03139, 2021.

[21] Maclagan, D., Rincón, F.: Tropical ideals. Compos. Math. 154 (2018), 3, 640-670. | DOI

[22] Maclagan, D., Rincón, F.: Tropical schemes, tropical cycles, and valuated matroids. J. Eur. Math. Soc. (JEMS) 22 (2020), 3, 777-796. | DOI

[23] Maclagan, D., Rincón, F.: Varieties of tropical ideals are balanced. arXiv:2009.14557, 2020.

[24] Maclagan, D., Sturmfels, B.: Introduction to tropical geometry. American Mathematical Society, Graduate Studies in Mathematics 161, Providence 2015.

[25] Macpherson, A. W.: Skeleta in non-Archimedean and tropical geometry. arXiv:1311.0502, 2013.

[26] Mikhalkin, G.: Tropical Geometry. Unfinish draft book.

[27] Mikhalkin, G.: Tropical geometry and its applications. In: International Congress of Mathematicians II, Eur. Math. Soc., Zürich 2006, pp. 827-852. | Zbl

[28] Mustata, M., Nicaise, J.: Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton. arXiv:1212.6328, 2013.

[29] Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16 (2009), 3, 543-556. | DOI

[30] Payne, S.: Fibers of tropicalization. Math. Z. 262 (2009), 2, 301-311. | DOI

[31] Popescu-Pampu, P., Stepanov, D.: Local tropicalization. In: Algebraic and combinatorial aspects of tropical geometry, Contemp. Math. 589, Amer. Math. Soc., Providence 2013, pp. 253-316. | DOI

[32] Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent mathematics and mathematical physics, Contemp. Math. 377, Amer. Math. Soc., Providence 2005, pp. 289-317. | DOI | Zbl

[33] Temkin, M.: Relative Riemann-Zariski spaces. Israel J. Math. 185 (2011), 1-42. | DOI

[34] Thuillier, A.: Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. applications à la théorie d'arakelov, 2005.

[35] Toën, B., Vaquié, M.: Au-dessous de ${\rm Spec} \mathbb{Z}$. J. K-Theory 3 (2009), 3, 437-500. | DOI

[36] Ulirsch, M.: Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Proc. Lond. Math. Soc. 3 114 (2017), 6, 1081-1113. | DOI

[37] Ulirsch, M.: Non-Archimedean geometry of Artin fans. Adv. Math. 345 (2019), 346-381. | DOI

[38] Włodarczyk, J.: Embeddings in toric varieties and prevarieties. J. Algebraic Geom. 2 (1993), 4, 705-726.

Cité par Sources :