Tautological cycles on tropical Jacobians

Gross, Andreas; Shokrieh, Farbod

doi:10.2140/ant.2023.17.885

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Tautological cycles on tropical Jacobians

Andreas Gross and Farbod Shokrieh

Vol. 17 (2023), No. 4, 885–921

DOI: 10.2140/ant.2023.17.885

Abstract

The classical Poincaré 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.

Keywords

Poincare formula, tropical curve, Jacobian, tropical homology

Mathematical Subject Classification

Primary: 14H40, 14H42, 14H51, 14T10

Milestones

Received: 20 October 2020

Revised: 1 May 2022

Accepted: 6 July 2022

Published: 2 May 2023

Authors

Andreas Gross
Institut für Mathematik Goethe-Universität Frankfurt Frankfurt Germany

Farbod Shokrieh
Department of Mathematics University of Washington Seattle, WA United States

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Vol. 17, No. 4, 2023