On the conditions for the parallelizability of a compact complex manifold

Abstract

In this note certain analytic and geometric conditions for the trivialization of a holomorphic vector bundle (on a compact complex space) am given. Applied to the holomorphic tangent bundle of a compact almost homogeneous complex manifold, these results yield parallelizability criteria for such manifolds. Especially,it is proved that a compaact, homogeneous, hermitian manifold with semi-negative scalar curvature is Ricci-flat and parallelizable.Similar results for manifolds admitting sufficiently many global holomorphic 1-forms are also obtained.