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.

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References in corpus (6)

• Skeletons of stable maps I: Rational curves in toric varieties
• Boundedness of the space of stable logarithmic maps
• Tropical geometry of moduli spaces of weighted stable curves
• Intersecting Psi-classes on tropical M_{0,n}
• On rational equivalence in tropical geometry
• Correspondence Theorems via Tropicalizations of Moduli Spaces

Cited by in corpus (6)

• A moduli stack of tropical curves
• Skeletons of stable maps I: Rational curves in toric varieties
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Moduli of stable maps in genus one and logarithmic geometry II
• Scattering diagrams, theta functions, and refined tropical curve counts
• The local/logarithmic correspondence and the degeneration formula for quasimaps