View source for Negative vector bundle

A holomorphic [[Vector bundle|vector bundle]] (cf. also [[Vector bundle, analytic|Vector bundle, analytic]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662501.png" /> over a [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662502.png" /> that possesses a [[Hermitian metric|Hermitian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662503.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662504.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662505.png" /> is strictly pseudo-convex outside the zero section (this is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662506.png" />). The vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662507.png" /> is negative if and only if the dual vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662508.png" /> (see [[Positive vector bundle|Positive vector bundle]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n0662509.png" /> is a manifold, then the condition of being negative can be expressed in terms of the curvature of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625010.png" />. Any subbundle of a negative vector bundle is negative. A vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625011.png" /> over a complex manifold is said to be negative in the sense of Nakano if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625012.png" /> is positive in the sense of Nakano. A holomorphic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625013.png" /> over a compact complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625014.png" /> is said to be weakly negative if its zero section possesses a strictly pseudo-convex neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625015.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625016.png" /> is weakly positive. Every negative vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625017.png" /> is weakly negative. Negative and weakly negative linear spaces over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066250/n06625018.png" /> are also defined in this way.

For references see [[Positive vector bundle|Positive vector bundle]].