On automorphism groups of fiber bundles
23 pages
English

On automorphism groups of fiber bundles

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23 pages
English
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Niveau: Supérieur, Doctorat, Bac+8
On automorphism groups of fiber bundles Michel Brion Abstract We obtain analogues of classical results on automorphism groups of holomorphic fiber bundles, in the setting of group schemes. Also, we establish a lifting property of the connected automorphism group, for torsors under abelian varieties. These results will be applied to the study of homogeneous bundles over abelian varieties. 1 Introduction This work arose from a study of homogeneous bundles over an abelian variety A, that is, of those principal bundles with base A and fiber an algebraic group G, that are isomorphic to all of their pull-backs by the translations of A (see [Br2]). In the process of that study, it became necessary to obtain algebro-geometric analogues of two classical results about automorphisms of fiber bundles in complex geometry. The first one, due to Morimoto (see [Mo]), asserts that the equivariant automorphism group of a principal bundle over a compact complex manifold, with fiber a complex Lie group, is a complex Lie group as well. The second one, a result of Blanchard (see [Bl]), states that a holomorphic action of a complex connected Lie group on the total space of a locally trivial fiber bundle of complex manifolds descends to a holomorphic action on the base, provided that the fiber is compact and connected.

  • fiber bundles

  • compact complex

  • scheme z

  • group scheme

  • fy ??

  • gaff

  • lie group

  • equivariant automorphisms


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Nombre de lectures 24
Langue English

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OnautomorphismgroupsoffiberbundlesMichelBrionAbstractWeobtainanaloguesofclassicalresultsonautomorphismgroupsofholomorphicfiberbundles,inthesettingofgroupschemes.Also,weestablishaliftingpropertyoftheconnectedautomorphismgroup,fortorsorsunderabelianvarieties.Theseresultswillbeappliedtothestudyofhomogeneousbundlesoverabelianvarieties.1IntroductionThisworkarosefromastudyofhomogeneousbundlesoveranabelianvarietyA,thatis,ofthoseprincipalbundleswithbaseAandfiberanalgebraicgroupG,thatareisomorphictoalloftheirpull-backsbythetranslationsofA(see[Br2]).Intheprocessofthatstudy,itbecamenecessarytoobtainalgebro-geometricanaloguesoftwoclassicalresultsaboutautomorphismsoffiberbundlesincomplexgeometry.Thefirstone,duetoMorimoto(see[Mo]),assertsthattheequivariantautomorphismgroupofaprincipalbundleoveracompactcomplexmanifold,withfiberacomplexLiegroup,isacomplexLiegroupaswell.Thesecondone,aresultofBlanchard(see[Bl]),statesthataholomorphicactionofacomplexconnectedLiegrouponthetotalspaceofalocallytrivialfiberbundleofcomplexmanifoldsdescendstoaholomorphicactiononthebase,providedthatthefiberiscompactandconnected.Also,weneededtoshowtheexistenceinthecategoryofschemesofcertainfiberbundlesassociatedtoaG-torsor(orprincipalbundles)π:XY,whereGisaconnectedgroupschemeandX,Yarealgebraicschemes;namely,thosefiberbundlesX×GZYassociatedtoG-homogeneousvarietiesZ.NotethatthefiberbundleassociatedtoanarbitraryG-schemeZexistsinthecategoryofalgebraicspaces,butmayfailtobeascheme(see[Bi,KM]).Finally,wewereledtoaliftingresultwhichreducesthestudyofhomogeneousbundlestothecasethatthestructuregroupislinear,anddoesnotseemtohaveitsholomorphic2010MathematicsSubjectClassification:14L10,14L15,14L30.1
counterpart.ItassertsthatgivenaG-torsorπ:XYwhereGisanabelianvarietyandX,Yaresmoothcompletealgebraicvarieties,theconnectedautomorphismgroupofXmapsontothatofYunderthehomomorphismprovidedbytheanalogueofBlanchard’stheorem.Inthispaper,wepresentthesepreliminaryresultswhichmayhaveindependentin-terest,with(hopefully)modestprerequisites.Section2isdevotedtoascheme-theoreticversionofBlanchard’stheorem:apropermorphismofschemesπ:XYsuchthatπ(OX)=OYinducesahomomorphismπ:Auto(X)Auto(Y)betweentheneu-tralcomponentsoftheautomorphismgroupschemes(Corollary2.2).Ourproofisanadaptationofthatgivenin[Ak]inthesettingofcomplexspaces.InSection3,weconsideratorsorπ:XYunderaconnectedgroupschemeG,andshowtheexistenceoftheassociatedfiberbundleX×GG/H=X/HforanysubgroupschemeHG(Theorem3.3).Asaconsequence,X×GZexistswhenZisthetotalspaceofaG-torsor,oragroupschemewhereGactsviaahomomorphism(Corollary3.4).AnotherapplicationofTheorem3.3concernsthequasi-projectivityoftorsors(Corollary3.5);itbuildsonworkofRaynaud,whoshowede.g.thelocalquasi-projectivityofhomogeneousspacesoveranormalscheme(see[Ra]).TheautomorphismgroupsoftorsorsarestudiedinSection4.Inparticular,weobtainaversionofMorimoto’stheorem:theequivariantautomorphismsofatorsoroveraproperschemeformagroupscheme,locallyoffinitetype(Theorem4.2).Hereourproof,basedonanequivariantcompletionofthestructuregroup,isquitedifferentfromtheoriginalone.Wealsoanalyzetherelativeequivariantautomorphismgroupofsuchatorsor;thisyieldsaversionofChevalley’sstructuretheoremforalgebraicgroupsinthatsetting(Proposition4.3).ThefinalSection5containsafulldescriptionofrelativeequivariantautomorphismsfortorsorsunderabelianvarieties(Proposition5.1)andourliftingresultforautomorphismsofthebase(Theorem5.4).Acknowledgements.ManythankstoGae¨lRe´mondforseveralclarifyingdiscussions,andspecialthankstotherefereeforveryhelpfulcommentsandcorrections.Infact,thefinalstepoftheproofofTheorem3.3istakenfromthereferee’sreport;theendoftheproofofCorollary2.2,andtheproofofCorollary3.4(ii),closelyfollowhis/hersuggestions.Notationandconventions.Throughoutthisarticle,weconsideralgebraicvarieties,schemes,andmorphismsoveranalgebraicallyclosedfieldk.Unlessexplicitlymentioned,wewillassumethattheconsideredschemesareoffinitetypeoverk(suchschemesarealsocalledalgebraicschemes).ByapointofaschemeX,wewillmeanaclosedpointunlessexplicitlymentioned.Avarietyisanintegralseparatedscheme.2
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