Tropical complexes

TL;DR

This paper introduces tropical complexes, enriching dual complexes with intersection data, and develops a theory of cycles, divisors, and intersections that bridges algebraic geometry and tropical geometry.

Contribution

It defines tropical complexes with intersection theory, connecting combinatorial structures to algebraic degenerations and extending previous tropical curve theories.

Findings

Defined cycles, divisors, and linear equivalence on tropical complexes

Established conditions for divisor-curve intersection numbers to match degenerations

Extended tropical geometry framework to include intersection data

Abstract

We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both to the corresponding theories on algebraic varieties and to previous work on graphs and abstract tropical curves. In addition, we establish conditions for the divisor-curve intersection numbers on a tropical complex to agree with the generic fiber of a degeneration.