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

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

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

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