View source for Vector bundle, analytic

<!--
v0964001.png
$#A+1 = 64 n = 0
$#C+1 = 64 : ~/encyclopedia/old_files/data/V096/V.0906400 Vector bundle, analytic
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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|Finiteness theorems]]). If, on the other hand,  $  X $
is a finite-dimensional complex [[Stein space|Stein space]], then  $  \Gamma ( E) $
is a [[Projective module|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 [[#References|[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|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)|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|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|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|Picard variety]] of the manifold  $  X $
and is thus a complex torus [[#References|[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 [[#References|[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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G. Fischer, "Lineare Faserräume und kohärente Modulgarben über komplexen Räumen" ''Arch. Math. (Basel)'' , '''18''' (1967) pp. 609–617 {{MR|0220972}} {{ZBL|0177.34402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Forster, K.J. Ramspott, "Über die Anzahl der Erzeugenden von projektiven Steinschen Moduln" ''Arch. Math. (Basel)'' , '''19''' (1968) pp. 417–422 {{MR|0236959}} {{ZBL|0162.38502}} </TD></TR></table>