Negative vector bundle

A holomorphic vector bundle (cf. also Vector bundle, analytic) $E$ over a complex space $X$ that possesses a Hermitian metric $h$ such that the function $v\to h(v,v)$ on $E$ is strictly pseudo-convex outside the zero section (this is denoted by $E<0$). The vector bundle $E$ is negative if and only if the dual vector bundle $E^*>0$ (see Positive vector bundle). If $X$ is a manifold, then the condition of being negative can be expressed in terms of the curvature of the metric $h$. Any subbundle of a negative vector bundle is negative. A vector bundle $E$ over a complex manifold is said to be negative in the sense of Nakano if $E^*$ is positive in the sense of Nakano. A holomorphic vector bundle $E$ over a compact complex space $X$ is said to be weakly negative if its zero section possesses a strictly pseudo-convex neighbourhood in $E$, i.e. if $E^*$ is weakly positive. Every negative vector bundle over $X$ is weakly negative. Negative and weakly negative linear spaces over a space $X$ are also defined in this way.