Uniform vector bundles on Fano manifolds and applications
type: Item
instance of: scholarly article
title: Uniform vector bundles on Fano manifolds and applications
zbMATH Open document ID: 1271.14058
DOI: 10.1515/CRELLE.2011.103
author: Roberto Muñoz
author: Gianluca Occhetta
author: Luis E. Solá Conde
publication date: 2012-04-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.2606
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6022597
zbMATH Keywords: vector bundle, uniform
zbMATH Keywords: splitting criteria
zbMATH Keywords: Fano bundle
zbMATH Keywords: Grassmannian
MaRDI profile type: Publication
arXiv ID: 0911.2606
cites work: The positivity of the Chern classes of an ample vector bundle
cites work: Stable rank 2 vector bundles with Chern-classes \(c_ 1=-1\), \(c_ 2=4\)
cites work: On the uniqueness of the Horrocks-Mumford-bundle
cites work: Caractérisation par l'uniformité des fibrés universels sur la grassmannienne
cites work: Stable vector bundles and instantons
cites work: The Chow ring of the Cayley plane
cites work: Self maps of homogeneous spaces
cites work: Uniform vector bundles on a projective space
OpenAlex ID: W2963458071
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published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(X\) be a Fano manifold with Picard number one admitting an unsplit covering family \(\mathcal M\) of rational curves. The authors prove a cohomological splitting criterion for uniform vector bundles on \(X\) of rank \(r \leq \dim \mathcal M_x\) (the family of curves of \(\mathcal M\) passing through a general point \(x \in X\)). If \(X\) embeds into a projective space in such a way that the curves in \(\mathcal M\) are lines, this criterion can be rephrased in terms of the variety of minimal rational tangents to \(X\) at \(x\). In particular, this allows the authors to recover many classical results about the splitting of uniform vector bundles of rank \(< n\) on \(\mathbb P^n\) and of rank \(\leq n-2\) (resp.\ \(n-3\)) on the smooth quadric \(\mathbb Q^n\) with \(n \geq 5\) odd (resp.\ even) and to apply their criterion to Grassmannians and other Hermitian symmetric spaces, determining the splitting treshold for uniform vector bundles in each case. Moreover, going one step further, they describe all non-splitting uniform vector bundles of lowest rank in several instances. Finally, the authors apply their results to the study of Fano bundles, namely, vector bundles \(E\) whose projectivizations are Fano manifolds, focusing on the case of rank-2 Fano bundles on Grassmannians. It turns out that any non-splitting Fano bundle of rank 2 on the Grassmannian of lines of \(\mathbb P^n\), \(n \geq 5\), is a twist of the universal quotient bundle.
reviewed by: Antonio Lanteri
description: scientific article; zbMATH DE number 6022597