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Ample vector bundles of small \(c_ 1\)-sectional genera. (English) Zbl 0699.14043

Let \({\mathcal E}\) be a vector bundle of rank r on a compact complex manifold M of dimension m and \(A=\det({\mathcal E})\). The \(c_ 1\)-sectional genus \(g=g(M,A)\) is defined by the relation \(2g(M,A)-2=(K+(m-1)A)A^{m-1}\) where K is the canonical bundle. The author classifies M and \({\mathcal E}\) for \(g=0,1,2.\)
If \(g=0\), \(M\cong P^ 2_{\alpha}\) and \({\mathcal E}=H_{\alpha}\oplus H_{\alpha}.\)
If \(g=1\), the classification breaks up into 5 similar parts.
When \(g=2\), the author uses the classification theory of polarized surfaces of sectional genus 2 to obtain a much more complicated 5 part classification broken up into subsections. Here, for instance, the first class contains M which are certain Jacobian varieties of smooth curves of genus 2, the second class (not known to exist) contains \(M\cong P({\mathcal F})\) with \({\mathcal F}\) a stable vector bundle of rank 2 and the fourth class consists of M which are the blowing-ups of \({\mathbb{P}}^ 2\) at 8 points.
Reviewer: P.Cherenack

MSC:

14J15 Moduli, classification: analytic theory; relations with modular forms
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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