Schubert polynomials and Arakelov theory of orthogonal flag varieties

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 𝔛= SO_N/B . We use these polynomials to describe the arithmetic Schubert calculus on 𝔛 . Moreover, we give a method to compute the natural arithmetic Chern numbers on 𝔛 , and show that they are all rational numbers.