Vector bundle, analytic

A locally trivial analytic bundle over an analytic space whose fibres have the structure of an $ n $-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