Representing a point and the diagonal as zero loci in flag manifolds

TL;DR

This paper explores how points and diagonals can be represented as zero loci of sections in vector bundles over flag manifolds, linking to Schubert polynomials and classical geometric problems.

Contribution

It provides a new perspective on realizing specific submanifolds, like points and diagonals, as zero loci in the context of flag manifolds, connecting to Schubert calculus.

Findings

Realization of points as zero loci in flag manifolds

Representation of diagonals as zero loci in product of flag manifolds

Connections to Schubert polynomials and classical geometry

Abstract

The zero locus of a generic section of a vector bundle over a manifold defines a submanifold. A classical problem in geometry asks to realise a specified submanifold in this way. We study two cases; a point in a generalised flag manifold and the diagonal in the direct product of two copies of a generalised flag manifold. These cases are particularly interesting since they are related to ordinary and equivariant Schubert polynomials respectively.