Continuity of the superpotentials and slices of tropical currents

TL;DR

This paper explores the continuity of slices of currents in tropical geometry, linking it to stable intersection theory and tropicalisation, and introduces new insights and techniques in tropical geometry and complex dynamics.

Contribution

It demonstrates that the continuity of superpotentials is a broad instance of stable intersection theory and connects tropicalisation properties to the behavior of currents.

Findings

Continuity of superpotentials relates to stable intersection theory.

Tropicalisation's properties influence the convergence of currents.

New methods for analyzing tropical currents and their slices.

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.