Balancing conditions in global tropical geometry

Abstract

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.

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© Association des Annales de l'institut Fourier, 2015. Attribution - Pas de Modification 3.0 France (CC BY-ND 3.0 FR) Manuscrit reçu le 17 juin 2013, révisé le 22 janvier 2015, accepté le 24 février 2015. I am very grateful to Maxim Kontsevich for inspiring discussions, from which this article originates. Discussions with Antoine Ducros, Pierrick Bousseau, Jean-François Dat, Ilia Itenberg, Sean Keel, Bernhard Keller, Bruno Klingler and Grigory Mikhalkin are equally very essential and useful. I would also like to thank the referees for valuable comments.

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Accepted Version - 1304.2251.pdf

Published - AIF_2015__65_4_1647_0.pdf

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