Negative vector bundle

A holomorphic vector bundle (cf. also Vector bundle, analytic) over a complex space that possesses a Hermitian metric such that the function on is strictly pseudo-convex outside the zero section (this is denoted by ). The vector bundle is negative if and only if the dual vector bundle (see Positive vector bundle). If is a manifold, then the condition of being negative can be expressed in terms of the curvature of the metric . Any subbundle of a negative vector bundle is negative. A vector bundle over a complex manifold is said to be negative in the sense of Nakano if is positive in the sense of Nakano. A holomorphic vector bundle over a compact complex space is said to be weakly negative if its zero section possesses a strictly pseudo-convex neighbourhood in , i.e. if is weakly positive. Every negative vector bundle over is weakly negative. Negative and weakly negative linear spaces over a space are also defined in this way.

For references see Positive vector bundle.