Balancing conditions in global tropical geometry

Yu, Tony Yue

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1647-1667.

Nous étudions la géométrie tropicale dans le cadre global en utilisant la rétraction par déformation construite par V. Berkovich. Nous montrons les conditions d’équilibre généralisées dans ce cadre. À partir d’un schéma formel strictement semi-stable, nous calculons certains faisceaux de cycles évanescents par la cohomologie étale analytique, puis nous interprétons les vecteurs de poids tropical via ces cycles. Nous obtenons la condition d’équilibre pour les courbes tropicales sur le squelette associé au schéma formel en fonction de la théorie d’intersection sur la fibre spéciale. Notre approche fonctionne pour tout corps complet de valuation discrète.

We study tropical geometry in the global setting using Berkovich’s deformation retraction. We state and prove the generalized balancing conditions in this setting. Starting with a strictly semi-stable formal scheme, we calculate certain sheaves of vanishing cycles using analytic étale cohomology, then we interpret the tropical weight vectors via these cycles. We obtain the balancing condition for tropical curves on the skeleton associated to the formal scheme in terms of the intersection theory on the special fiber. Our approach works over any complete discrete valuation field.

Classification : 14T05, 14G22
Keywords: balancing condition, tropical curve, Berkovich space, vanishing cycle
Mot clés : condition d’équilibre, courbe tropicale, espace de Berkovich, cycle évanescent

[1] Atiyah, M. F. On analytic surfaces with double points, Proc. Roy. Soc. London. Ser. A, Volume 247 (1958), pp. 237-244 | DOI | MR | Zbl

[2] Baker, Matthew; Payne, Sam; Rabinoff, Joseph Nonarchimedean geometry, tropicalization, and metrics on curves (2011) (http://arxiv.org/abs/1104.0320)

[3] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990, pp. x+169 | MR | Zbl

[4] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. (1993) no. 78, p. 5-161 (1994) | DOI | EuDML | Numdam | MR | Zbl

[5] Berkovich, Vladimir G. Vanishing cycles for formal schemes, Invent. Math., Volume 115 (1994) no. 3, pp. 539-571 | DOI | EuDML | MR | Zbl

[6] Berkovich, Vladimir G. Vanishing cycles for formal schemes. II, Invent. Math., Volume 125 (1996) no. 2, pp. 367-390 | DOI | MR | Zbl

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

[8] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Singular semipositive metrics in non-archimedean geometry (2011) (http://arxiv.org/abs/1201.0187)

[9] Dat, Jean-François A lemma on nearby cycles and its application to the tame Lubin-Tate space, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 165-173 | DOI | MR | Zbl

[10] Ducros, Antoine La structure des courbes analytiques (En cours de rédaction, date du 15/11/2012)

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

[12] Grothendieck, Alexander Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973, pp. x+438 Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz

[13] Gubler, Walter A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry (Contemp. Math.), Volume 589, Amer. Math. Soc., Providence, RI, 2013, pp. 125-189 | DOI | MR

[14] Gubler, Walter; Rabinoff, Joseph; Werner, Annette Skeletons and tropicalizations (2014) (http://arxiv.org/abs/1404.7044)

[15] Kontsevich, Maxim; Tschinkel, Yury Non-archimedean Kähler geometry (2002) (www.ihes.fr/~maxim/TEXTS/Non-archimedean Kahler geometry.pdf)

[16] Matsusaka, T. The criteria for algebraic equivalence and the torsion group, Amer. J. Math., Volume 79 (1957), pp. 53-66 | DOI | MR | Zbl

[17] Mikhalkin, Grigory Enumerative tropical algebraic geometry in ${ℝ}^2$, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 313-377 | DOI | MR | Zbl

[18] Nishinou, Takeo; Siebert, Bernd Toric degenerations of toric varieties and tropical curves, Duke Math. J., Volume 135 (2006) no. 1, pp. 1-51 | DOI | MR | Zbl

[19] Rapoport, M.; Zink, Th. Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., Volume 68 (1982) no. 1, pp. 21-101 | DOI | MR | Zbl

[20] Speyer, David E Uniformizing tropical curves I: genus zero and one (2007) (http://arxiv.org/abs/0711.2677)

[21] Yu, Tony Yue Gromov compactness in non-archimedean analytic geometry (2014) (http://arxiv.org/abs/1401.6452)