Intersection Theory on Tropicalizations of Toroidal Embeddings

TL;DR

This paper develops an intersection theory on tropicalizations of toroidal embeddings, establishing a framework that links tropical and algebraic geometry through multiplicities and cycle correspondences.

Contribution

It introduces a new intersection theory on cone complexes of toroidal embeddings, connecting tropical cycles with algebraic cycles via multiplicities and push-forward compatibility.

Findings

Defined a balancing condition for weighted subcomplexes.

Established a correspondence between tropical and algebraic Gromov-Witten invariants.

Proved that tropicalization maps respect intersections and rational equivalence.

Abstract

We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, intersections with boundary divisors, and rational equivalence. As an application we prove a correspondence between the genus 0 tropical descendant Gromov-Witten invariants introduced by Markwig and Rau and the genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.