Continuity of the superpotentials and slices of tropical currents

2025/06/11 by Farhad Babaee, Babaee, Farhad, Tien Cuong Dinh +1
Computer Science · Mathematics · #14T10 #32U40 #37F80 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2506.09828

openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the question of the continuity of slices of currents and explain how it relates to several seemingly unrelated problems in tropical geometry. On the one hand, through this lens, we show that the continuity of superpotentials constitutes a very general instance of stable intersection theory in tropical geometry. On the other hand, questions concerning tropicalisation with respect to a non-trivial valuation and the commutativity of tropicalisation with intersections offer insights to the problem of the continuity of slices of currents converging to a tropical current. Within our framework, we reformulate and reprove known theorems in tropical geometry and derive new results and techniques in both tropical geometry and complex dynamical systems.

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Related

• Super currents and tropical geometry
• Complex Tropical Currents, Extremality, and Approximations
• On Tropical Intersection Theory
• A guide to tropical modifications
• Tropical Images of Intersection Points