A new mathematical preprint has narrowed the boundary for a broad class of vector bundles on generalized Grassmannians. For any generalized Grassmannian that is not projective space, it shows that the largest rank guaranteed to produce a homogeneous bundle is no more than one below the space’s dimension. On odd-dimensional quadrics, that boundary is exact: k(Q^(2n-1)) = 2n-2.
Here k(X) is a threshold: the largest integer for which every uniform bundle of rank at most k(X) is homogeneous, or governed by the same full symmetry as the underlying space. The paper’s main objects are uniform vector bundles on n-dimensional smooth quadric hypersurfaces Qn, studied to determine k(Qn). In practical terms, the work asks how long that guarantee of symmetry lasts before non-homogeneous bundles can enter the picture.
A clear answer for odd dimensions
That distinction is clearest in the quadric results. For an odd-dimensional quadric Q^(2n-1), the threshold is k = 2n-2. For an even-dimensional quadric Q^(2n), the preprint gives only 2n-2 ≤ k ≤ 2n-1. It therefore narrows the answer to two neighboring possibilities but does not identify a single exact value in even dimensions.
The classification becomes more concrete for rank n in the cases n = 3 and n = 5. In each of those dimensions, every uniform bundle on Qn, up to duality, either splits or takes one of the listed spinor-plus-line, tangent-twist or orthogonal-kernel forms.
Uniform does not always mean homogeneous
An orthogonal kernel bundle supplies one of the paper’s central examples. On every quadric Qn with n at least 3, the bundle is uniform with splitting type (0, ..., 0, -1). The same family is non-homogeneous and has rank n. In other words, it meets the paper’s uniformity condition without meeting the stronger symmetry condition measured by homogeneity.
The authors also isolate a narrower case on odd quadrics. For n at least 3, every uniform bundle of rank 2n-1 on Q^(2n-1) with splitting type (0, ..., 0, -1) either splits or is an orthogonal kernel bundle. The qualification matters: this is not a complete classification of every uniform bundle of rank 2n-1 on an odd quadric, and that broader classification remains open.
A broader construction
Beyond the quadric examples, an iterative quotient construction produces a globally generated, uniform bundle E of rank n with splitting type (1, 0, ..., 0). Whenever the generalized Grassmannian is not projective space, every n-bundle from the stated construction is non-homogeneous. The construction therefore gives the paper a way to exhibit uniform bundles outside the homogeneous class.
How the classifications were obtained
The classification arguments use exact sequences induced by relative Harder–Narasimhan filtrations. The proofs also calculate Chern classes using Whitney’s formula and solve the resulting polynomial systems with Mathematica; routine algebraic details are omitted.
What remains unresolved
Several questions remain. The exact threshold for even-dimensional quadrics is still not fixed, and the full rank-(2n-1) classification on odd quadrics remains open beyond the specified splitting type. The work thus sets a sharp odd-quadric result and supplies non-homogeneous examples while leaving those broader classification problems unresolved.
A criterion for projective space
The paper ends with a broader characterization. A generalized Grassmannian of dimension n at least 2 is projective space exactly when its tangent bundle is the unique unsplit uniform bundle of minimal rank, allowing duality and twist. This links the geometry of the space to the behavior of its most basic bundle.
Publication details
The front matter identifies the work as arXiv:2608.25921v1, dated 26 August 2026. Xinyi Fang reports support from the National Natural Science Foundation of China under Grant Nos. 12501057 and 12471040, while Yuhang Zhou reports support from the CAS Project for Young Scientists in Basic Research under Grant No. YSBR-032.
Paper data and sources
Original title: Uniform non-homogeneous bundles on quadrics
Authors: Xinyi Fang, Yuhang Zhou
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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