Morphisms from $\mathbb{P}^m$ to flag varieties

TL;DR

This paper investigates morphisms from projective spaces to flag varieties, establishing conditions under which such morphisms are constant and deriving implications for the splitting types of certain vector bundles.

Contribution

It proves that non-constant morphisms from projective spaces to flag varieties only exist under specific conditions, and applies this to restrict possible splitting types of uniform vector bundles.

Findings

Morphisms from $P^m$ to flag varieties are mostly constant under certain conditions.

Certain splitting types of uniform $r$-bundles on $P^m$ are impossible.

Provides new constraints on vector bundle structures over projective spaces.

Abstract

In this paper, we consider the morphisms from projective spaces to flag varieties. We show that the morphisms can only be constant under some special conditions. As a consequence, we prove that the splitting types of unsplit uniform $r$-bundles on $\mathbb{P}^m$ can not be $(a_1,\dots,a_1,a_2,\dots,a_{r-k+1})$ for $1\le k\le m-2$.