View source for Principal analytic fibration

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746401.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746403.png" /> are analytic spaces (cf. [[Analytic space|Analytic space]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746404.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746405.png" /> is an analytic mapping; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746406.png" /> is a Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746407.png" /> analytically acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746408.png" /> from the right; and for each element of the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p0746409.png" /> there exist a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464010.png" /> and an analytic isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464011.png" /></td> </tr></table>

such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464012.png" /></td> </tr></table>

Each analytic vector bundle (cf. [[Vector bundle, analytic|Vector bundle, analytic]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464014.png" />-dimensional fibre determines a principal analytic fibration with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464015.png" /> and group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464016.png" />, whose fibre over a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464017.png" /> is the variety of all bases of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464018.png" />. This is a special case of a one-to-one correspondence between analytic fibrations with given fibre and structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464019.png" /> and the principal analytic fibrations associated with them. Other examples of principal analytic fibrations include the fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464020.png" /> whose fibres are the left cosets of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464021.png" /> by a Lie subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464023.png" /> of its base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464024.png" /> and transition functions, i.e. analytic mappings

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464025.png" /></td> </tr></table>

which satisfy the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464026.png" /></td> </tr></table>

The transition functions form a one-dimensional cocycle with values in the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464027.png" /> of germs of analytic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464028.png" />. This yields a one-to-one correspondence between the set of principal analytic fibrations with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464029.png" /> and group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464030.png" /> and the set of cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464031.png" />. The classification of the principal analytic fibrations with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464032.png" /> and structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464033.png" /> (in the case of a non-Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464034.png" />) 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464035.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464036.png" /> is the sheaf of germs of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464037.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464038.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464039.png" /> be a reduced [[Stein space|Stein space]]; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464041.png" />-principal fibrations, in the definition of which the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464042.png" /> is replaced by some locally trivial analytic fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464043.png" /> on complex Lie groups, acting fibre-wise on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464044.png" />. 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464045.png" /> is a compact Riemann surface, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464046.png" /> is surjective and not injective. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464047.png" /> need not be either surjective nor injective. Sufficient cohomological conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464048.png" /> to be injective or surjective are known [[#References|[3]]] for a connected solvable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464049.png" />.

The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464050.png" /> is a compact Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464052.png" /> is a connected reductive algebraic group has been also studied. See [[#References|[5]]] for the corresponding classification for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464053.png" />; see [[#References|[1]]] for the corresponding classification for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464054.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464055.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464057.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464058.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464059.png" /> and over algebraic surfaces has also been obtained.

See [[Real-analytic space|Real-analytic space]] for real principal analytic fibrations (the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074640/p07464060.png" />).

====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 sphere 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>