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Numerically effective vector bundles with small Chern classes. (English) Zbl 0781.14006
Complex algebraic varieties, Proc. Conf., Bayreuth/Ger. 1990, Lect. Notes Math. 1507, 145-156 (1992).
[For the entire collection see Zbl 0745.00049.]
A vector bundle $$E$$ over a projective variety $$X$$ is called nef (numerically effective) if the relative very ample line bundle $$L$$ on $$\mathbb{P}(E)$$, which is $${\mathcal O}(1)$$ on fibres and has direct image $$E$$ on $$X$$, is nef. If $$X$$ is the projective space $$\mathbb{P}^ n$$ or a smooth quadric $$Q_ n$$, it is easy to see that any nef vector bundle $$E$$ with $$c_ 1(E)=0$$ is trivial. The authors give a complete classification of nef vector bundles with $$c_ 1(E)=1, 2$$ on $$\mathbb{P}^ n$$ and with $$c_ 1(E)=1$$ on $$Q_ n$$. The classification for lower $$n$$’s done by M. Szurek and J. A. Wisniewski [Pac. J. Math. 141, No. 1, 197-208 (1990; Zbl 0705.14016) and Nagoya Math. J. 120, 89-101 (1990; Zbl 0728.14037)] is used here. The Stein factorisation $$X\to Y$$ of the morphism given by the complete linear system $$| L|$$ is in fact a contraction of an extremal ray for the Fano manifold $$\mathbb{P}(E)$$ for $$c_ 1(E) \leq 2$$. Study of this contraction morphism coupled with the standard vector bundle techniques leads to the classification of nef vector bundles in this paper.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J45 Fano varieties 57R20 Characteristic classes and numbers in differential topology 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli