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Difference between revisions of "Negative vector bundle"

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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.
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A holomorphic [[Vector bundle|vector bundle]] (cf. also [[Vector bundle, analytic|Vector bundle, analytic]]) $E$ over a [[Complex space|complex space]] $X$ that possesses a [[Hermitian metric|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|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.
  
 
For references see [[Positive vector bundle|Positive vector bundle]].
 
For references see [[Positive vector bundle|Positive vector bundle]].

Latest revision as of 14:21, 10 August 2014

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.

For references see Positive vector bundle.

How to Cite This Entry:
Negative vector bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_vector_bundle&oldid=15576
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article