Uniform bundles on quadrics
Xinyi Fang, Duo Li, Yanjie Li
TL;DR
The paper investigates uniform vector bundles on generalized Grassmannians, establishing splitting results and structure theorems via a geometric approach tied to VMRTs and via an approximate-solution framework for Chern-polynomial constraints. It proves that uniform bundles of rank $2n$ on the quadrics $\mathbb{Q}^{2n+1}$ and $\mathbb{Q}^{2n+2}$ split for $n\ge3$, showing the splitting threshold $\mu(\mathbb{Q}^{2n+1})=2n$, and providing a first example where this threshold exceeds the e.d.(VMRT). Through a detailed analysis of morphisms from quadrics to Grassmannians, it derives conditions under which such morphisms must be constant, and extends these insights to related homogeneous spaces. The work also classifies unsplit uniform bundles of minimal rank on certain $B_n/P_k$ and $D_n/P_k$, and offers partial progress toward Ellia's conjecture by constraining splitting types of unsplit uniform bundles. Collectively, these results deepen the understanding of splitting phenomena for uniform bundles on generalized Grassmannians and connect geometric invariants with algebraic splitting behavior.
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.