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As far as I know, it is still unknown whether there exists a (holomorphic) indecomposable vector bundle of rank $r$ on $\mathbb{P}^n_{\mathbb{C}}$ with $n\geq 6$ and $1<r< n-1$. What is the situation for hypersurfaces? Does anyone know a smooth hypersurface in $\mathbb{P}^{n+1}_{\mathbb{C}}$ ($n> 6$) carrying an indecomposable vector bundle of rank $r$ with $1<r< n-1$?

Edit: As pointed out by Dragon, I overlooked the spinor bundles on 6-dimensional quadrics, so let me assume $n>6$.

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What about the paper by GIORGIO OTTAVIANI on spinor bundles on quadrics? There is a rank 4 spinor bundle on a dimension 6 quadric, unless I am mistaken?

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