Tropical intersection homology

TL;DR

This paper introduces a tropical analog of intersection homology to extend the geometric understanding of algebraic cycles and their numerical equivalence beyond classical cases.

Contribution

It generalizes tropical cohomology to pairs of smooth proper varieties and divisors, creating a new framework for intersection homology in tropical geometry.

Findings

Provides a new tropical intersection homology theory for pairs of varieties and divisors.

Connects tropical geometry with classical intersection theory in algebraic geometry.

Lays groundwork for further exploration of tropical analogs of classical invariants.

Abstract

Numerical equivalence of algebraic cycles is defined abstractly by intersection numbers. Classically, for smooth complex proper toric varieties, the quotients by numerical equivalence with rational coefficients can be described geometrically as singular cohomology. They are also expressed in terms of tropical geometry, tropical cohomology, introduced by Itenberg-Katzarkov-Mikhalkin-Zharkov. This paper aims to generalize this to suitable pairs of smooth proper varieties and divisors by introducing a tropical analog of intersection homology.