Uniform vector bundles of rank \((n+1)\) on \({\mathbb{P}}_ n\). (English) Zbl 0532.14006

Let E be a rank-r vector bundle over \({\mathbb{P}}^ n={\mathbb{P}}^ n(K)\), \(ch(K)=0\). E is said to be uniform if there exists a sequence of integers \((k;r_ 1,...,r_ k;a_ 1,...,a_ k)\) such that for every line L of \({\mathbb{P}}^ n\), \(E| L\cong \oplus^{k}_{i=1}{\mathcal O}_ L(a_ i)\). For low r there are only a few uniform vector bundles of rank r on \({\mathbb{P}}^ n\). E. Sato [J. Math. Soc. Japan 28, 123-132 (1976; Zbl 0315.14003)] and G. Elencwajg, A. Hirschowitz and M. Schneider [cf. Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 37-63 (1980; Zbl 0456.32009)] proved that for \(r\leq n\) every uniform bundle is a direct sum of line bundles, \(\Omega_{{\mathbb{P}}^ n}(a)\), \(T{\mathbb{P}}^ n(b)\). Here I prove that this holds for \(r=n+1\). For many n (e.g. if \(n+1\) is prime) this result was previously proved by Ph. Ellia in Mem. Soc. Math. Fr., Nouv. Sér. 7 (1982; Zbl 0505.14015. I use Ellia’s tricks and methods: the Harder- Narasimhan filtration used by Elencwajg-Hirschowitz-Schneider.


14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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