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.

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.

Reviewer: U.N.Bhosle (Bombay)

##### 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 |