A Tropical Framework for Using Porteous Formula

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. We also discuss the foundational considerations that necessitated the adoption of the Mikhalkin--Rau framework, explaining why the standard R^n-based framework of Allermann--Rau is insufficient for a meaningful theory of degeneracy loci.