Tautological cycles on tropical Jacobians

TL;DR

This paper establishes a tropical analogue of the classical Poincaré formula for tautological cycles on Jacobians, along with foundational results on tropical abelian varieties and cycle equivalences.

Contribution

It introduces a tropical version of the Poincaré formula and proves foundational results like the tropical Appell-Humbert theorem, advancing tropical geometry theory.

Findings

Tropical Poincaré formula for tautological cycles

Tropical Appell-Humbert theorem established

Analysis of cycle equivalences in tropical setting

Abstract

The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several foundational results about real tori with integral structures (and, therefore, tropical abelian varieties). For example, we prove a tropical version of the Appell-Humbert theorem. We also study various notions of equivalences between tropical cycles and their relation to one another.