Uniform bundles on quadrics

TL;DR

This paper investigates the structure and splitting properties of uniform bundles on quadrics and related homogeneous varieties, establishing new splitting criteria, classifying minimal rank bundles, and partially confirming a conjecture on bundle splitting.

Contribution

It proves the non-existence of certain constant morphisms, improves splitting bounds for uniform bundles, classifies minimal rank bundles on specific homogeneous spaces, and addresses a conjecture on splitting types.

Findings

Only constant morphisms exist from certain quadrics to Grassmannians when conditions are met.

Uniform bundles of rank at most 2m on certain quadrics split, improving previous bounds.

Classified all minimal rank uniform bundles on specific homogeneous varieties.

Abstract

We show that there exist only constant morphisms from $\mathbb{Q}^{2n+1}(n\geq 1)$ to $\mathbb{G}(l,2n+1)$ if $l$ is even $(0<l<2n)$ and $(l,2n+1)$ is not $ (2,5)$. As an application, we prove on $\mathbb{Q}^{2m+1}$ and $\mathbb{Q}^{2m+2}(m\geq 3)$, any uniform bundle of rank at most $2m$ splits, which improves the upper bound of splitting for uniform bundles obtained by Kachi and Sato. We classify all unsplit uniform bundles of minimal rank on $B_n/P_k$ $(k=\frac{2n}{3},k\ge6)$ and $D_n/P_k$ $(k=\frac{2n-2}{3},k\ge 6)$. We partially answer a conjecture of Ellia, which predicts that some uniform bundles of special splitting types on $\mathbb{P}^n$ necessarily split and we find some restrictions on the splitting types of unsplit uniform bundles of minimal rank.