Nonexistence of regular maps between homogeneous projective varieties

TL;DR

This paper proves that there are no non-constant regular maps from partial flag varieties to complete flag varieties, and proposes a broader conjecture on the non-existence of such maps between homogeneous projective varieties.

Contribution

It establishes non-existence results for regular maps between certain homogeneous varieties and introduces a general conjecture in this area.

Findings

No non-constant regular maps from partial to complete flag varieties.

Formulation of a general conjecture on non-existence of regular maps.

Provides a theoretical framework for understanding morphisms between homogeneous varieties.

Abstract

We study non-existence of non-constant regular maps from a partial flag variety $X=G/P$ to another partial flag variety $X'=G'/P'$ and prove that there does not exist any non-constant regular map from any partial non-complete flag variety $X$ to any complete flag variety $X'=G'/B'$. We also formulate a general conjecture on the non-existence of non-constant regular maps from $X\to X'$.