Morphisms between Grassmannians, II

TL;DR

The paper establishes a rigidity result for morphisms between Grassmannians by showing that any nonconstant map $\varphi:\mathbb G(l,n) \to \mathbb G(k,n)$ with $l \neq 0,n-1$ must satisfy $l=k$ or $l=n-k-1$, and in those cases $\varphi$ is an isomorphism. The proof combines effective good divisibility, a full classification of maximal disjoint pairs (md-pairs) in the Grassmannian Chow rings, and Schubert-calculus arguments, culminating in a Remmert–Van de Ven type result in the setting of rational homogeneous varieties (Hwang–Mok). This work extends and sharpens previous results (Tango1, Tango2, NO22, MOS8) by providing a precise dichotomy and rigidity for endomorphisms of the same Grassmannian, with implications for the structure of morphisms in the homogeneous setting. The methods yield a clear criterion based on cycle intersection properties that precludes nontrivial maps unless the domains and targets align up to a duality, thereby establishing strong geometric constraints on possible morphisms.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Definition 2.1
• Example 2.2
• Theorem 2.3: Cf. MOS8, HLL
• Definition 2.4
• Example 2.5
• Example 2.6
• Proposition 2.7
• Example 2.8
• Theorem 3.1