Balancing conditions in global tropical geometry

TL;DR

This paper extends tropical geometry to a global setting using Berkovich's deformation retraction, establishing generalized balancing conditions for tropical curves via intersection theory and sheaves of vanishing cycles.

Contribution

It introduces a new approach to global tropical geometry by connecting Berkovich's deformation retraction with intersection theory and sheaves of vanishing cycles.

Findings

Generalized balancing conditions for tropical curves on the skeleton.

Calculation of sheaves of vanishing cycles using analytic étale cohomology.

Balancing conditions expressed in terms of intersection theory on the special fiber.

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 \'etale cohomology, then we interpret the tropical weights 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.