On ample and spanned vector bundles with zero \(\Delta\)-genera. (English) Zbl 0728.14012

Let E be an ample spanned vector bundle of rank two over a complex manifold S with \(\dim (S)=n=2k\) even. The author defines the degree \(d(E):=(c_ 1(E)^ 2-c_ 2(E))c_ 2(E)^{k-1}\) and the \(\Delta\)- genus \(\Delta (E):=k+2+d(E)-h^ 0(E)\) of E. He then proves d(E)\(\geq 3\) and \(\Delta\) (E)\(\geq 0\). Moreover, if either \(d(E)=3\) or \(\Delta (E)=0\), then \(S\cong {\mathbb{P}}^ n\) and \(E={\mathcal O}(1)\oplus {\mathcal O}(1)\). In the proof, he reduces the problem to the case \(n=2\) by Bertini technique, where the theory of \(\Delta\)-genus of polarized manifolds applies, since \(d(E)=d(P,H)\) and \(\Delta (E)=\Delta (P,H)\) for the tautological line bundle H on the threefold \(P={\mathbb{P}}_ S(E)\).
Reviewer: T.Fujita (Tokyo)


14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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