## 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)

### MSC:

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

### Keywords:

ample spanned vector bundle; degree; $$\Delta$$ -genus
Full Text:

### References:

 [1] [B1] E. Ballico:Spanned and ample vector bundles with low Chern numbers, Pacific J. of Math.140 (1989), 209–216 · Zbl 0638.14010 [2] [B2] E. Ballico:On vector bundles on threefolds with sectional genus 1, Trans. A.M.S. (to appear) [3] [B3] E. Ballico:On ample and spanned rank-3 bundles with low Chern numbers, Manuscripta math.68 (1990), 9–16 · Zbl 0717.14011 [4] [BG] S. Bloch–D. Gieseker:The positivity of the Chern classes of an ample vector bundle, Invent. math.12 (1971), 112–117 · Zbl 0212.53502 [5] [F] T. Fujita:On polarized varieties of small {$$\Delta$$}-genera, Tohoku Math. J.34 (1982), 319–341 · Zbl 0489.14002 [6] [FL] W. Fulton–R. Lazarsfeld:Positive polynomials for ample vector bundles, Ann. Math.118 (1983), 35–60 · Zbl 0537.14009 [7] [G] D. Gieseker:p-ample bundles and their Chern classes, Nagoya math. J.43 (1971), 91–116 [8] [L] R. Lazarsfeld:Some applications of the theory of positive vector bundles, in: Complete intersections–Acireale 1983, pp. 29–61, Springer Lect. Notes in Math.1092 (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.