Curvature and linear subbundles of holomorphic vector bundles

Abstract

We study the behavior of holomorphic vector bundles with supplementary curvature conditions. Let L ~ M be a holomorphic line or bundle on the complex manifold M and g be a fibrewise Hermitian metric of positive curvature on L. Then (as follows easily from [i, p. 170]) the function g(7, ~) has no relative minima for any everywhere nonzero section ~ ~ H ° (M~ ~ (L)). Generalizing this fact, S. Kobayashi noted [2, p. 166] that a global holomorphic section of a holomorphic Hermitian bundle with positive curvature and compact base has a nonempty set of zeros (the outline of the corresponding argument is given in [2]). Unfortunately, the latter is not always true (cf. the example below). Kobayashi's remark is valid under an additional restriction.