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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.

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