View source for Principal analytic fibration

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A locally trivial analytic fibration on the fibres of which the structure Lie group acts simply transitively and analytically. In other words, a principal analytic fibration is a quadruple  $  (P,\  B,\  G,\  \pi ) $
where  $  P $
and  $  B $
are analytic spaces (cf. [[Analytic space|Analytic space]]) over a field  $  k $; 
$  \pi : \  P \rightarrow B $
is an analytic mapping;  $  G $
is a Lie group over  $  k $
analytically acting on  $  P $
from the right; and for each element of the base  $  B $
there exist a neighbourhood  $  U $
and an analytic isomorphism $$ 
\phi : \  U \times G   \rightarrow   \pi ^{-1} (U)
$$
such that $$ 
\phi (x,\  gh)   =   \phi (x,\  g) h,  
x \in U,   g,\  h \in G.
$$
Each analytic vector bundle (cf. [[Vector bundle, analytic|Vector bundle, analytic]])  $  p: \  V \rightarrow B $
with  $  n $-
dimensional fibre determines a principal analytic fibration with base  $  B $
and group  $   \mathop{\rm GL}\nolimits (n,\  k) $, 
whose fibre over a point  $  b \in B $
is the variety of all bases of the fibre  $  p ^{-1} (b) $. 
This is a special case of a one-to-one correspondence between analytic fibrations with given fibre and structure group  $  G $
and the principal analytic fibrations associated with them. Other examples of principal analytic fibrations include the fibration  $  H \rightarrow H/G $
whose fibres are the left cosets of the Lie group  $  H $
by a Lie subgroup  $  G $
of it, and an analytic [[Covering|covering]] (here, the group of the covering is the structure group).

A principal analytic fibration can be defined by an open covering  $  {\{ U _{i} \}} _{ {i} \in I} $
of its base  $  B $
and transition functions, i.e. analytic mappings $$ 
g _{ij} : \  U _{i} \cap U _{j}   \rightarrow   G,  
i,\  j \in I,  
U _{i} \cap U _{j} \neq \emptyset ,
$$
which satisfy the conditions $$ 
g _{ij} (x) g _{jk} (x)   =   g _{ik} (x),  
x \in U _{i} \cap U _{j} \cap U _{k} .
$$
The transition functions form a one-dimensional cocycle with values in the sheaf  $  {\mathcal O} ^{G} $
of germs of analytic mappings  $  B \rightarrow G $. 
This yields a one-to-one correspondence between the set of principal analytic fibrations with base  $  B $
and group  $  G $
and the set of cohomology spaces  $  H ^{1} (B,\  {\mathcal O} ^{G} ) $. 
The classification of the principal analytic fibrations with base  $  B $
and structure group  $  G $(
in the case of a non-Abelian group  $  G $) 
is a problem which has so far been solved in a number of cases only (cf. [[Deformation|Deformation]] of an analytic structure, for the corresponding local moduli problem). The connection between this classification and the classification of principal topological fibrations is expressed by the natural mapping $$ 
\alpha : \  H ^{1} (B,\  {\mathcal O} ^{G} )   \rightarrow   H ^{1} (B,\  C ^{G} ),
$$
where  $  C ^{G} $
is the sheaf of germs of continuous mappings  $  B \rightarrow G $.


Let  $  k = \mathbf C $, 
and let  $  B $
be a reduced [[Stein space|Stein space]]; the mapping  $  \alpha $
is then bijective, and the classification problem is reduced to a problem in homotopy theory [[#References|[4]]]. In particular, any principal analytic fibration with a non-compact Riemann surface as base and with a connected structure group is trivial. The result of [[#References|[4]]] was in fact obtained for the broader class of  $  E $-
principal fibrations, in the definition of which the group  $  G $
is replaced by some locally trivial analytic fibration  $  E \rightarrow B $
on complex Lie groups, acting fibre-wise on  $  P $. 
This result has been generalized to the case when the structure group is a complex Banach Lie group [[#References|[2]]]. If, on the other hand,  $  B $
is a compact Riemann surface, then  $  \alpha $
is surjective and not injective. In the general case  $  \alpha $
need not be either surjective nor injective. Sufficient cohomological conditions for  $  \alpha $
to be injective or surjective are known [[#References|[3]]] for a connected solvable group  $  G $.


The case when  $  B $
is a compact Riemann surface of genus  $  p $
and  $  G $
is a connected reductive algebraic group has been also studied. See [[#References|[5]]] for the corresponding classification for  $  p = 0 $; 
see [[#References|[1]]] for the corresponding classification for  $  p = 1 $
when  $  G =  \mathop{\rm GL}\nolimits (n,\  \mathbf C ) $. 
In the case  $  p \geq 2 $, 
$  G =  \mathop{\rm GL}\nolimits (n,\  \mathbf C ) $(
cf. [[Vector bundle, algebraic|Vector bundle, algebraic]]) an important role is played by the concept of a stable vector bundle. This concept has been generalized [[#References|[6]]] to the case of an arbitrary connected reductive group  $  G $. 
The set of all stable principal fibrations of a given topological type on a compact Riemann surface has the natural structure of a connected [[Normal analytic space|normal analytic space]]. A partial classification of vector bundles over the complex projective space  $  B = P ^{m} ( \mathbf C ) $
and over algebraic surfaces has also been obtained.

See [[Real-analytic space|Real-analytic space]] for real principal analytic fibrations (the case  $  k = \mathbf R $).


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> M.F. Atiyah, "Vector bundles over an elliptic curve" ''Proc. London Math. Soc. (3)'' , '''7''' (1957) pp. 414–452 {{MR|0131423}} {{ZBL|0084.17305}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Bungart, "On analytic fibre bundles - I. Holomorphic fibre bundles with infinite dimensional fibres" ''Topology'' , '''7''' (1968) pp. 55–68 {{MR|0222338}} {{MR|0201673}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Frenkel, "Cohomologie non Abélienne et espaces fibrés" ''Bull. Soc. Math. France'' , '''85''' (1957) pp. 135–220 {{MR|0098200}} {{ZBL|0082.37702}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Grauert, "Analytische Faserungen über holomorph-vollständigen Räumen" ''Math. Ann.'' , '''135''' (1958) pp. 263–273 {{MR|0098199}} {{ZBL|0081.07401}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A. Grothendieck, "Sur la classification des fibres holomorphes sur la sphère de Riemann" ''Amer. J. Math.'' , '''79''' (1957) pp. 121–138 {{MR|0087176}} {{ZBL|0079.17001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Ramanathan, "Stable principal bundles on a compact Riemann surface" ''Math. Ann.'' , '''213''' (1975) pp. 129–152 {{MR|0369747}} {{ZBL|0284.32019}} </TD></TR>
</table>

====Comments====
For the partial classification of bundles over a complex projective space or an algebraic surface see [[#References|[a1]]]–[[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. van de Ven, "Twenty years of classifying algebraic vectorbundles" A. Beauville (ed.) , ''Journées de géometrie algébrique d'Angers'' , Sijthoff &amp; Noordhoff (1980) pp. 3–20 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Harshorne, "Four years of algebraic vectorbundles" A. Beauville (ed.) , ''Journées de géometrie algébrique d'Angers (1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 21–29 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> C. Okonek, M. Schneider, H. Spindler, "Vector bundles on complex projective spaces" , Birkhäuser (1980) {{MR|0561910}} {{ZBL|0438.32016}} </TD></TR></table>