Schubert polynomials and Arakelov theory of orthogonal flag varieties

Abstract

We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of the orthogonal flag variety \({\mathfrak X={\rm SO}_N/B}\). We use these polynomials to describe the arithmetic Schubert calculus on \({\mathfrak X}\). Moreover, we give a method to compute the natural arithmetic Chern numbers on \({\mathfrak X}\), and show that they are all rational numbers.