Skeleta in non-Archimedean and tropical geometry
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 431-506.

Je décris une théorie algèbro-géometrique de squelettes, qui fournit une cadre unifiée pour l’étude de variétés tropicaux, les squelettes des variétés analytiques non Archimediennes, et les variétés différentielles avec structure affine singulier. Ces squelettes sont des espaces munies d’un faisceau structural de semianneaux topologiques, et sont localement isomorphes aux spectres de ceux-ci. Le résultat principal de cet article dit que l’espace topologique sous-jacent d’une variété analytique non-Archimedienne peut être localement reconstruit par les sections du faisceau de valuations “point-par-point” de ses fonctions analytiques.

I describe an algebro-geometric theory of skeleta, which provides a unified setting for the study of tropical varieties, skeleta of non-Archimedean analytic spaces, and affine manifolds with singularities. Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same. The primary result of this paper is that the topological space X underlying a non-Archimedean analytic space may locally be recovered from the sections of the sheaf |𝒪 X | of pointwise valuations of its analytic functions; in other words, (X,|𝒪 X |) is a skeleton.

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DOI : 10.5802/afst.1637
Licence : CC-BY 4.0
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     title = {Skeleta in {non-Archimedean} and tropical geometry},
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Andrew W. Macpherson. Skeleta in non-Archimedean and tropical geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 2, pp. 431-506. doi : 10.5802/afst.1637. https://afst.centre-mersenne.org/articles/10.5802/afst.1637/

[1] Michael Artin; Alexander Grothendieck; Jean-Louis Verdier Théorie des topos et cohomologie étale des schémas. (SGA 4), Lecture Notes in Mathematics, Springer, 1972 | Zbl

[2] Matthew Baker; Sam Payne; Joseph Rabinoff Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom., Volume 3 (2016) no. 1, pp. 63-105 | DOI | MR | Zbl

[3] Vladimir G. Berkovich Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci. (1993) no. 78, pp. 5-161 | DOI | Numdam | MR | Zbl

[4] Vladimir G. Berkovich Smooth p-adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl

[5] Nicolas Bourbaki Séminaire Banach, Springer, 1962

[6] Nikolai Durov A new approach to Arakelov geometry (2007) (https://arxiv.org/abs/0704.2030) | Zbl

[7] Manfred Einsiedler; Mikhail Kapranov; Douglas Lind Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math., Volume 601 (2006), pp. 139-157 | MR | Zbl

[8] Kazuhiro Fujiwara; Fumiharu Kato Foundations of rigid geometry. I, EMS Monographs in Mathematics, European Mathematical Society, 2018, xxxiv+829 pages | Zbl

[9] Jeffrey Giansiracusa; Noah Giansiracusa Equations of tropical varieties, Duke Math. J., Volume 165 (2016) no. 18, pp. 3379-3433 | DOI | MR | Zbl

[10] Mark Gross Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, 114, American Mathematical Society, 2011, xvi+317 pages | MR | Zbl

[11] Mark Gross; Bernd Siebert From real affine geometry to complex geometry, Ann. Math., Volume 174 (2011) no. 3, pp. 1301-1428 | DOI | MR | Zbl

[12] Alexander Grothendieck; Jean A. Dieudonné Eléments de géométrie algébrique. I, Grundlehren der Mathematischen Wissenschaften, 166, Springer, 1971, ix+466 pages | Zbl

[13] Roland Huber Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30, Vieweg & Sohn, 1996, x+450 pages | MR | Zbl

[14] Maxim Kontsevich; Yan Soibelman Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2001, pp. 203-263 | DOI | Zbl

[15] Maxim Kontsevich; Yan Soibelman Affine structures and non-Archimedean analytic spaces, The unity of mathematics (Progress in Mathematics), Volume 244, Birkhäuser, 2006, pp. 321-385 | DOI | MR | Zbl

[16] Saunders Mac Lane; Ieke Moerdijk Sheaves in geometry and logic, Universitext, Springer, 1994, xii+629 pages (A first introduction to topos theory, Corrected reprint of the 1992 edition) | Zbl

[17] Grigory Mikhalkin Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 827-852 | Zbl

[18] Johannes Nicaise; Chenyang Xu The essential skeleton of a degeneration of algebraic varieties, Am. J. Math., Volume 138 (2016) no. 6, pp. 1645-1667 | DOI | MR | Zbl

[19] Sam Payne Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 3, pp. 543-556 | DOI | MR | Zbl

[20] Bertrand Toën; Michel Vaquié Au-dessous de Spec , J. K-Theory, Volume 3 (2009) no. 3, pp. 437-500 | DOI | MR | Zbl

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