Intersection theory on tropicalizations of toroidal embeddings

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.