Positive vector bundle
A generalization of the concept of a divisor of positive degree on a Riemann surface. A holomorphic vector bundle over a complex space
is called positive (denoted by
) if on
there exists a Hermitian metric
such that the function
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on is strictly pseudo-convex outside the zero section. If
is a manifold, the condition for positivity is expressed in terms of the curvature of the metric
. That is, the curvature form of the metric
in the bundle
corresponds to a Hermitian quadratic form
on
with values in the bundle
of Hermitian endomorphisms of the bundle
. The positivity condition is equivalent to
being a positive-definite operator on
for any
and any non-zero
.
If is a complex line bundle over a manifold
, the condition for positivity is equivalent to that of positive definiteness of the matrix
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where are local coordinates in
and
is a function that defines the Hermitian metric for the local trivialization of the bundle. If
is compact, then the complex line bundle
over
is positive if and only if the Chern class
contains a closed form of the type
![]() |
where is a positive-definite Hermitian matrix. In particular, if
is a Riemann surface, then the bundle over
defined by a divisor of degree
is positive if and only if
. If
is a bundle of rank
over a manifold
of dimension
, one can consider also the following narrower class of positive bundles: A bundle
is called positive in the sense of Nakano if there exists on
a Hermitian metric
such that the Hermitian quadratic form
on the bundle
as defined by the formula
![]() |
where ,
and
, is positive definite. Examples: the tangent bundle
of the projective space
is positive, but for
it is not positive in the sense of Nakano; the complex line bundle over
defined by a hyperplane is positive.
Any quotient bundle of a positive vector bundle is positive. If and
are positive bundles (in the sense of Nakano), then
and
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) 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 over a compact complex space
is called weakly negative if its zero section has a strictly pseudo-convex neighbourhood in
, i.e. if it is an exceptional analytic set. A bundle
is called weakly positive if the dual bundle
is weakly negative. If
is a Riemann surface, the concepts of a weakly-positive bundle and a positive bundle coincide [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 is equivalent to each of the following properties: For any coherent analytic sheaf
on
there exists an
such that
for
is generated by global sections; for any analytic sheaf
on
there exists an
such that
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for all [3], [4]. By
is meant the sheaf of germs of holomorphic sections of the bundle
. Weakly-positive bundles are therefore analogous to the ample sheafs (cf. Ample sheaf) from algebraic geometry and are sometimes called ample analytic bundles. A weakly-positive bundle over a space
naturally defines an imbedding of
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 (see Vector bundle, analytic).
See also Negative vector bundle.
References
[1] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004 |
[2] | S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004 |
[3] | 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 |
[4] | M. Schneider, "Familien negativer Vektorraumbündel und 1-konvexe Abbildungen" Abh. Math. Sem. Univ. Hamburg , 47 (1978) pp. 150–170 MR0492393 Zbl 0391.32011 |
[5] | H. Umemura, "Some results in the theory of vector bundles" Nagoya Math. J. , 52 (1973) pp. 97–128 MR0337968 Zbl 0271.14005 Zbl 0253.14003 |
Positive vector bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_vector_bundle&oldid=12720