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

TL;DR

The paper addresses when morphisms from $\mathbb{P}^m$ to flag varieties are nonconstant, using Chow ring computations and Schubert-cycle positivity to prove constancy in several regimes, and constructs nonconstant maps in odd dimensions via a fibered approach. It connects these morphism rigidity results to the classification of uniform vector bundles on $\mathbb{P}^m$, showing that a uniform bundle with a splitting type containing $k$ identical largest parts (with $1\le k\le m-2$) must split into line bundles. The main contributions include precise constancy results for maps to flag varieties $A_n/P(I)$ when $\Delta(A_n)\setminus I$ is a consecutive block, a nonconstant-map criterion in odd dimensions, and a new splitting theorem for uniform bundles, highlighting the interplay between morphisms, Schubert calculus, and bundle theory. These results advance understanding of rigidity phenomena in projective geometry and have potential implications for the classification of vector bundles on 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 $1\le k\le m-2$.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: See guyot1985caracterisation Theorem 3.2
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3