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