Morphisms between Grassmannians, II

TL;DR

This paper proves that non-constant morphisms between Grassmannians of certain dimensions are either isomorphisms or trivial, specifically when the dimensions are not at the extremes, revealing a rigidity property of Grassmannian morphisms.

Contribution

It establishes a classification of morphisms between Grassmannians, showing that under specific conditions such morphisms are either isomorphisms or trivial, extending previous understanding of Grassmannian mappings.

Findings

Non-constant morphisms occur only when dimensions match or are complementary.

Such morphisms are either isomorphisms or trivial.

The result applies to Grassmannians with dimensions not at the boundary cases.

Abstract

Denote by $\mathbb G(k,n)$ the Grassmannian of linear subspaces of dimension $k$ in $\mathbb P^n$. We show that, if $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ is a non constant morphism and $l \not=0,n-1$ then $l=k$ or $l=n-k-1$ and $\varphi$ is an isomorphism.