Splitting conjectures for uniform flag bundles
Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde
TL;DR
The paper addresses the problem of diagonalizability for uniform $G/B$-bundles on rational homogeneous spaces by formulating splitting conjectures that compare Coxeter-type invariants ${\mathop{\rm h}(X)}$ and ${\mathop{\rm r}(X)}$ with those of the bundle ${\pi:Y\to X}$ via ${\mathop{\rm h}(\pi)}$ and ${\mathop{\rm r}(\pi)}$. It builds a framework around flag bundles, parabolic reductions, and the cohomology of rational homogeneous varieties to derive obstructions and criteria for diagonalizability, including a criterion based on uniform reducibility along families of rational curves. The main results establish the splitting conjecture ${\mathop{\rm h}}$ for classical groups and ${\mathop{\rm r}}$ for type ${\rm A}_n$, with a detailed analysis of morphisms to homogeneous spaces and the resulting cohomological constraints. These findings connect uniformity to homogeneity in a broad flag-bundle setting, offering concrete criteria and partial evidence toward a full understanding of when uniform flag bundles split as diagonalizable objects.
Abstract
We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the validity of these conjectures in the case of classical groups.