Schubert polynomials and Arakelov theory of symplectic flag varieties

TL;DR

This paper develops explicit combinatorial Schubert polynomials for symplectic flag varieties and applies them to arithmetic Schubert calculus, enabling the computation of rational arithmetic Chern numbers.

Contribution

It introduces a new combinatorial framework for Schubert polynomials in symplectic geometry and connects it to arithmetic intersection theory.

Findings

Schubert classes are represented by explicit polynomials

Arithmetic Chern numbers on symplectic flag varieties are rational

Method for computing arithmetic Chern numbers is established

Abstract

Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. 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.