Morphisms from projective spaces to flags of minimal parabolic subgroups

TL;DR

This paper classifies morphisms from projective spaces to flag varieties of SL(n,C), showing nonexistence results for low-dimensional cases and providing elementary proofs using cohomology.

Contribution

It provides a complete classification of morphisms from projective spaces to flag varieties for SL(n,C) up to dimension 4, correcting previous errors and employing elementary cohomological methods.

Findings

No nonconstant morphism from P^2 to G/B.

No nonconstant morphisms from P^3 to certain G/P_{α_i} for i in {1, n-1}.

No nonconstant morphism from P^4 to any G/P_{α_i}.

Abstract

We classify nonconstant morphisms $\mathbb{P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we show that there is no nonconstant morphism $\mathbb{P}^2 \to G/B$; for minimal parabolic subgroup $P_{\alpha_i}$, there are no nonconstant morphisms $\mathbb{P}^3 \to G/P_{\alpha_i}$ when $i \in \{1, n-1\}$, while such morphisms exist for $1 < i < n-1$; and, after correcting an earlier error (pointed out by Yanjie Li), we give an elementary proof that there is no nonconstant morphism $\mathbb{P}^4 \to G/P_{\alpha_i}$ for any minimal parabolic subgroup. The proofs are elementary and cohomological.