Morphisms from projective spaces to flags of minimal parabolic subgroups

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.

arXiv commentIn this version we add a corrected proof of the nonexistence of nonconstant morphism ${\mathbb{P}}^4\to G/P_{{\alpha }_i}$ for any minimal parabolic subgroup