View source for Positive vector bundle

A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphic [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739801.png" /> over a [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739802.png" /> is called positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739803.png" />) if on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739804.png" /> there exists a [[Hermitian metric|Hermitian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739805.png" /> such that the function

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on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739807.png" /> is strictly pseudo-convex outside the zero section. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739808.png" /> is a manifold, the condition for positivity is expressed in terms of the curvature of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739809.png" />. That is, the [[Curvature form|curvature form]] of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398010.png" /> in the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398011.png" /> corresponds to a Hermitian quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398013.png" /> with values in the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398014.png" /> of Hermitian endomorphisms of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398015.png" />. The positivity condition is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398016.png" /> being a positive-definite operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398017.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398018.png" /> and any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398019.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398020.png" /> is a complex line bundle over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398021.png" />, the condition for positivity is equivalent to that of positive definiteness of the matrix

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398023.png" /> are local coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398025.png" /> is a function that defines the Hermitian metric for the local trivialization of the bundle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398026.png" /> is compact, then the complex line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398027.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398028.png" /> is positive if and only if the [[Chern class|Chern class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398029.png" /> contains a closed form of the type

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398031.png" /> is a positive-definite Hermitian matrix. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398032.png" /> is a Riemann surface, then the bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398033.png" /> defined by a divisor of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398034.png" /> is positive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398036.png" /> is a bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398037.png" /> over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398038.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398039.png" />, one can consider also the following narrower class of positive bundles: A bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398040.png" /> is called positive in the sense of Nakano if there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398041.png" /> a Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398042.png" /> such that the Hermitian quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398043.png" /> on the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398044.png" /> as defined by the formula

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398048.png" />, is positive definite. Examples: the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398049.png" /> of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398050.png" /> is positive, but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398051.png" /> it is not positive in the sense of Nakano; the complex line bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398052.png" /> defined by a hyperplane is positive.

Any quotient bundle of a positive vector bundle is positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398054.png" /> are positive bundles (in the sense of Nakano), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398056.png" /> are positive (in the sense of Nakano).

The concept of a positive bundle was introduced in connection with Kodaira's vanishing theorem (cf. [[Kodaira theorem|Kodaira theorem]]) for complex line bundles, and it was then extended to any bundle. Somewhat later, in relation to the existence of an imbedding in a projective space, the concepts of weakly-positive and weakly-negative bundles were introduced.

A holomorphic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398057.png" /> over a compact complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398058.png" /> is called weakly negative if its zero section has a strictly pseudo-convex neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398059.png" />, i.e. if it is an [[Exceptional analytic set|exceptional analytic set]]. A bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398060.png" /> is called weakly positive if the dual bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398061.png" /> is weakly negative. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398062.png" /> is a Riemann surface, the concepts of a weakly-positive bundle and a positive bundle coincide [[#References|[5]]]. In the general case, positivity implies weak positivity; no examples are known at present (1983) of weakly positive but non-positive bundles.

Weak positivity of a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398063.png" /> is equivalent to each of the following properties: For any [[Coherent analytic sheaf|coherent analytic sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398065.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398066.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398067.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398068.png" /> is generated by global sections; for any analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398069.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398070.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398071.png" /> such that

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for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398073.png" /> [[#References|[3]]], [[#References|[4]]]. By <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398074.png" /> is meant the sheaf of germs of holomorphic sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398075.png" />. Weakly-positive bundles are therefore analogous to the ample sheafs (cf. [[Ample sheaf|Ample sheaf]]) from algebraic geometry and are sometimes called ample analytic bundles. A weakly-positive bundle over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398076.png" /> naturally defines an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398077.png" /> into a Grassmann manifold and thus into a projective space.

The concepts of a positive, a negative, a weakly-positive, and a weakly-negative bundle are naturally extended also to the case of a linear space over a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398078.png" /> (see [[Vector bundle, analytic|Vector bundle, analytic]]).

See also [[Negative vector bundle|Negative vector bundle]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Chern,   "Complex manifolds without potential theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.L. Onischik,   "Pseudoconvexity in the theory of complex spaces"  ''J. Soviet Math.'' , '''14''' :  4  (1980)  pp. 1363–1406  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''15'''  (1977)  pp. 93–171</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Schneider,   "Familien negativer Vektorraumbündel und 1-konvexe Abbildungen"  ''Abh. Math. Sem. Univ. Hamburg'' , '''47'''  (1978)  pp. 150–170</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Umemura,   "Some results in the theory of vector bundles"  ''Nagoya Math. J.'' , '''52'''  (1973)  pp. 97–128</TD></TR></table>