Skeleta in non-Archimedean and tropical geometry

Macpherson, Andrew W.

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, $(X,|{𝒪}_X\left|\right)$ is a skeleton.

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

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

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

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

[5] Bourbaki, Nicolas Séminaire Banach, Springer, 1962

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

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

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

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

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

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

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

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

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

[15] Kontsevich, Maxim; Soibelman, Yan 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] Mac Lane, Saunders; Moerdijk, Ieke 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] Mikhalkin, Grigory Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 827-852 | Zbl

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

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

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