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Fano bundles and splitting theorems on projective spaces and quadrics. (English) Zbl 0808.14013
Here the authors study the structure of Fano bundles (i.e., projectivizations of vector bundles with ample anticanonical divisor) in dimension \(\geq 4\) using vector bundles techniques and Mori theory. Their main results are the following ones.
Theorem 1. Let \(E\) be a rank 2 Fano bundle on \(\mathbb{C} \mathbb{P}^ n\) or the hyperquadric \(Q_ n\). If \(n \geq 4\), up to some explicit completely described examples on \(Q_ 4\) and \(Q_ 5\), \(E\) splits into the direct sum of two line bundles.
Note that if rank\((E) = 2\), then \(\mathbb{P}(E)\) is a Fano manifold on \(\mathbb{C} \mathbb{P}^ n\) or the hyperquadric \(Q_ n\) if and only if, up to a twist, \(E\) is ample with \(c_ 1 (E) \leq n + 1\).
Theorem 2. Let \(F\) be an ample rank 2 vector bundle on \(\mathbb{C} \mathbb{P}^ n\); \(F\) splits if either \(n=4\) and \(c_ 1(F) \leq 6\) or \(n=5\) and \(c_ 1(F) \leq 8\) or \(6 \leq n \leq 7\) and \(c_ 1(F) \leq (4n + 2)/3\) or \(n \geq 8\) and \(c_ 1(F) \leq (5n - 1)/3\).
The authors give other splitting criteria for vector bundles and criteria for having the same Chern classes as a splitted bundle. On this topic, see also the reviewer, “A splitting criterion for rank 2 vector bundles on \(\mathbb{P}^ n\)”. [Pac. J. Math. (to appear)].
Reviewer: E.Ballico (Povo)

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J40 \(n\)-folds (\(n>4\))
14J45 Fano varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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