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Vector bundle, analytic

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A locally trivial analytic bundle over an analytic space whose fibres have the structure of an -dimensional vector space over a ground field k (if k = \mathbf C is the field of complex numbers, the analytic bundle is said to be holomorphic). The number n is said to be the dimension, or rank, of the bundle. Similarly as for a topological bundle (cf. Vector bundle), definitions are given of the category of analytic vector bundles, and of the concepts of a subbundle, a quotient bundle, the direct sum, the tensor product, the exterior product of analytic vector bundles, etc.

The analytic sections of an analytic vector bundle with base X form a module \Gamma ( E) over the algebra A( X) of analytic functions on the base. If k = \mathbf C and X is compact, \Gamma ( E) is a finite-dimensional vector space over \mathbf C (see Finiteness theorems). If, on the other hand, X is a finite-dimensional complex Stein space, then \Gamma ( E) is a projective module of finite type over A( X) , and the correspondence E \mapsto \Gamma ( E) defines an equivalence between the category of analytic vector bundles over X and the category of projective A( X) -modules of finite type [4].

Examples of analytic vector bundles include the tangent bundle of an analytic manifold X (its analytic sections are analytic vector fields on X ), and the normal bundle of a submanifold Y \subset X .

The classification of analytic vector bundles of rank n on a given analytic space X is equivalent with the classification of principal analytic fibrations (cf. Principal analytic fibration) with base X and structure group \mathop{\rm GL} ( n, k) and, for n > 1 , has been completed only in certain special cases. For projective complex algebraic varieties X it is identical with the classification of algebraic vector bundles (cf. Comparison theorem (algebraic geometry)).

Analytic vector bundles of rank 1 on a complex space X (in other words, bundles of complex lines, or line bundles) play an important part in complex analytic geometry. Each divisor on the space X necessarily defines an analytic bundle of rank 1, two divisors defining isomorphic bundles if and only if they are linearly equivalent. All analytic line bundles on a projective algebraic variety are defined by a divisor. The imbeddability of a complex space X into a projective space is closely connected with the existence of ample line bundles on X (cf. Ample vector bundle). If one is given a discrete group \Gamma of automorphisms of a complex space X , each quotient of \Gamma will determine a line bundle over X/ \Gamma , with the respective automorphic forms as its analytic sections. Analytic vector bundles of rank 1 constitute the group H ^ {1} ( X, {\mathcal O} _ {X} ^ {*} ) , where {\mathcal O} _ {X} ^ {*} is the sheaf of invertible elements of the structure sheaf. The correspondence between each bundle and its first Chern class yields the homomorphism

\gamma : H ^ {1} ( X, {\mathcal O} _ {X} ^ {*} ) \rightarrow H ^ {2} ( X, \mathbf Z ),

whose kernel is the set of topologically trivial line bundles. If X is a complex manifold, \mathop{\rm Im} \gamma may be described as the set of cohomology classes which are representable by closed differential forms of type ( 1, 1) . If, in addition, X is compact and Kählerian, \mathop{\rm Ker} \gamma is isomorphic to the Picard variety of the manifold X and is thus a complex torus [2].

To each analytic vector bundle V of rank n on an analytic space X corresponds a sheaf of germs of analytic sections of V , which is a locally free analytic sheaf of rank n on X . This correspondence defines an equivalence between the categories of analytic vector bundles and locally free analytic sheaves on X . Attempts to generalize this result to arbitrary coherent analytic sheaves resulted in the following generalization of the concept of an analytic vector bundle [3]: A surjective morphism \pi : V \rightarrow X is said to be an analytic family of vector spaces over X (or a linear space over X ) if its fibres have the structure of finite-dimensional vector spaces over k , and if the operations of addition, multiplication by a scalar and the zero section satisfy the natural conditions of analyticity. If k= \mathbf C (or k= \mathbf R and X is coherent), the analytic family of vector spaces \pi : V \rightarrow X defines a coherent analytic sheaf F on X : For U \subset X the group F( U) is the space of analytic functions on \pi ^ {-1} ( U) which are linear on the fibres. In the same way is defined the duality between the categories of analytic families of vector spaces and coherent analytic sheaves on X .

References

[1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[2] S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004
[3] G. Fischer, "Lineare Faserräume und kohärente Modulgarben über komplexen Räumen" Arch. Math. (Basel) , 18 (1967) pp. 609–617 MR0220972 Zbl 0177.34402
[4] O. Forster, K.J. Ramspott, "Über die Anzahl der Erzeugenden von projektiven Steinschen Moduln" Arch. Math. (Basel) , 19 (1968) pp. 417–422 MR0236959 Zbl 0162.38502
How to Cite This Entry:
Vector bundle, analytic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_bundle,_analytic&oldid=52087
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article