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.

A graphical interface for the Gromov-witten theory of curves

“…The approach of logarithmic Gromov-Witten theory is more flexible, generalizes immediately to higher dimensional target toric varieties, and allows one to study invariants with psi class insertions. See [20,36]. .…”

“…Proof. As in the previous lemma, we express the left hand side of (20) as an integral over a space of rubber stable maps [32]. Again, we adopt the convention that in rubber relative stable maps all relative points are marked, which allows us to absorb the automorphism factors,…”

Descendant log Gromov-Witten invariants for toric varieties and tropical curves

“…Concerning genus 0 log invariants, while our paper was in progress, [Ran17] used ideas from tropical intersection theory to give a new proof of the Nishinou-Siebert result, and then A. Gross [Gro18] extended these methods to allow for gravitational ancestors (i.e., pullbacks of ψ-classes from M 0,n , which by our Prop. 3.4 turn out to coincide with the usual descendant ψ-classes).…”

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

“…Let be a Zariski logarithmic scheme that is logarithmically smooth over the base field. In , Gross has developed a version of tropical intersection theory on (extended) cone complexes that admit a weak embedding into a vector space generated by a lattice, expanding on the theory developed in this article. In particular, using his approach one can enrich the tropicalization map to an operation on algebraic cycles on that nontrivially intersect .…”