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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
Bănică sheaf
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