A tropical framework for using Porteous formula

TL;DR

This paper develops a tropical analogue of Porteous' formula by defining characteristic classes of tropical vector bundles on rational polyhedral spaces with boundary, enabling determinantal expressions for degeneracy loci.

Contribution

It introduces a framework for characteristic classes of tropical vector bundles with boundary and establishes a tropical Porteous' formula, extending classical degeneracy locus theory to tropical geometry.

Findings

Defined tropical characteristic classes for vector bundles with boundary

Proved a tropical splitting principle

Established a determinantal formula for degeneracy loci

Abstract

Given a rational polyhedral space $X$ (a tropical cycle with boundary, in the sense of Mikhalkin--Rau), one can define tropical vector bundles on $X$ having real or tropical fibers. By restricting attention to bounded rational sections of these bundles, one obtains characteristic classes that behave as expected classically. We develop further properties of these classes and use them to prove a tropical analogue of the splitting principle, which allows us to establish the foundations for Porteous' formula in this setting: a determinantal expression for the fundamental class of degeneracy loci in terms of Chern classes. The boundary framework is essential, as it allows the rank of a bundle morphism to drop at sedentary strata, giving degeneracy loci their expected codimension.