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Uniform vector bundles on Fano manifolds and applications. (English) Zbl 1271.14058
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.

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J45 Fano varieties
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