Ample vector bundles of small \(c_ 1\)-sectional genera (Q912939)

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.