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On Bănică sheaves and Fano manifolds. (English) Zbl 0872.14009
Definition: Let \({\mathcal E}\) be a coherent sheaf of rank \(r\geq 2\) over a normal variety \(Y\) and let \(p: \mathbb{P} ({\mathcal E}) \to Y\) be the projectivization of \({\mathcal E}\). Then \({\mathcal E}\) is called a Bănică sheaf if \(\mathbb{P} ({\mathcal E})\) is smooth.
Consequently, the authors show that all Bănică sheaves are reflexive. – Next, they show that
(a) \(Y\) is smooth and \({\mathcal E}\) is locally free if \(r\geq\dim Y\) and
(b) \(Y\) is smooth if \(r= \dim Y-1\).
Also examples of Bănică sheaves which are not locally free are exhibited.
In section 4, cohomological obstructions for Bănică sheaves of rank \(n-1\) over a smooth variety \(Y\), to be extensions of locally free sheaves are established.
Finally, the classification of \({\mathcal E}\), \(Y\) and \(\mathbb{P} ({\mathcal E})\) is given for non locally free \({\mathcal E}\) and \(\mathbb{P} ({\mathcal E})\) which is assumed to be a Fano manifold of index \(r=\frac12 \dim\mathbb{P} ({\mathcal E})\).

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J45 Fano varieties
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
Keywords:
Bănică sheaf
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