Curvature and linear subbundles of holomorphic vector bundles (Q923171)

\textit{S. Kobayashi} stated without proof [Nagoya Math. J. 57, 153-166 (1975; Zbl 0326.32016)] that a global holomorphic section of a Hermitian holomorphic vector bundle E with positive curvature over a compact complex manifold M has always a zero point. The author first presents an example showing that this assertion is not true. Nevertheless he proves that this assertion is true under the additional assumption fiberdim \(E\leq \dim M\). More precisely he proves the following theorem: Let \(E\to M\) be a holomorphic vector bundle with fiberdim \(E\leq \dim M\), and let h be a Hermitian metric on E with positive curvature. Then for any nowhere vanishing holomorphic section \(\gamma\) of E the function h(\(\gamma\),\({\bar \gamma}\)) attains at no point its local minimum. Two interesting corollaries of the theorem follow.