Balancing conditions in global tropical geometry
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1647-1667.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2970
Classification: 14T05,  14G22
Keywords: balancing condition, tropical curve, Berkovich space, vanishing cycle
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Yu, Tony Yue. Balancing conditions in global tropical geometry. Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1647-1667. doi : 10.5802/aif.2970. https://aif.centre-mersenne.org/articles/10.5802/aif.2970/

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