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A tropical framework for using Porteous formula

Andrew R. Tawfeek

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.

A tropical framework for using Porteous formula

Abstract

Given a rational polyhedral space (a tropical cycle with boundary, in the sense of Mikhalkin--Rau), one can define tropical vector bundles on 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.

Paper Structure

This paper contains 19 sections, 29 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Tropical Preliminaries
  4. Rational Polyhedral Spaces
  5. Tropical Intersection Theory
  6. Tropical Vector Bundles
  7. Tropical Linear Algebra
  8. Tropical Vector Bundles
  9. Tropical Chern Classes
  10. Tropical Splitting Principle
  11. Tropical projectivization
  12. Porteous formula
  13. Sylvester determinants
  14. Degeneracy Loci
  15. The Hom bundle
  16. ...and 4 more sections

Key Result

Theorem 1.0.1

Let $\varphi: \mathcal{E} \to \mathcal{F}$ be a morphism of vector bundles of ranks $e$ and $f$ on a smooth variety $X$. If the scheme $D_k(\varphi) \subseteq X$ has codimension $(e-k)(f-k)$, then its class is given by

Figures (3)

  • Figure 1: An abstract tropical polyhedral complex, as well as a covering with open fans. Here, $H_f \subseteq \mathbb{R}^2$ denotes the fan having cones $\{ f(x) \leq 0\}$, $\{f(x) = 0 \}$, and $\{f(x) \geq 0 \}$.
  • Figure 2: The tropical projective line $\mathbb{T}\mathbb{P}^1$, with rays emanating from the origin analogous to the classical projective line.
  • Figure 3: The pullback of $\mathcal{O}_{\mathbb{T}\mathbb{P}^2}(1)$ along an embedding of a curve into ${\mathbb{T}\mathbb{P}}^2$.

Theorems & Definitions (96)

  • Theorem 1.0.1: Porteous Formula
  • Theorem 1.0.2: Tropical Porteous Formula in Rank $0$
  • Definition 2.1.1: Sedentarity
  • Definition 2.1.2: Rational polyhedral complex in $\mathbb{T}^n$
  • Remark 1
  • Definition 2.2.1: Abstract tropical cycles
  • Definition 2.2.2: Abstract tropical subcycles
  • Definition 2.2.3: Rational functions
  • Definition 2.2.4: Regular invertible functions
  • Remark 2
  • ...and 86 more