关于Riemann曲面上Ample向量丛的一个注记

A Remark on Ampleness of Vector Bundles Over Riemann Surfaces

本文证明如果E是亏格为P的紧Riemann曲面R上秩为r的不可约向量丛,deg(E)>r(r-1)(P-1),则E是Ample向量丛。并且对于任意正整数r,存在R上秩为r的不可约向量丛E_r,使得deg(E_r)=r(r-1)(P-1),但E_r不是Ample的。

Let R be a compact Riemannian surface of genus P,E an irreducible holomorphic vector bundle of rank r over R.It is proved that if deg(E)>r(r- 1)(P- 1), then E is an ample vector bundle.It is also shown that for each integer r there exists an irreducible vector bundle Er of rank r with deg(Er) = r(r - 1) (P- 1), but Er is not ample.