Schubert polynomials and Arakelov theory of orthogonal flag varieties

TL;DR

This paper develops explicit Schubert polynomials for orthogonal flag varieties and applies them to arithmetic intersection theory, providing a combinatorial and computational framework for arithmetic Schubert calculus.

Contribution

It introduces a combinatorially explicit theory of Schubert polynomials for orthogonal flag varieties and applies it to arithmetic intersection calculations.

Findings

Schubert classes are represented by explicit polynomials

Arithmetic intersection numbers are rational

Method for computing intersection numbers is provided

Abstract

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic Schubert calculus on these spaces. We also give a method to compute the natural arithmetic intersection numbers which arise, and prove that they are all rational numbers.