Classical Symmetries of Complex Manifolds 9 0 0 Alan Huckleberry &AlexanderIsaev 2 n a J 7 Abstract 2 We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such ] V manifolds in a natural situation. Under our assumptions, which require C the groupathand tobe dimension-theoreticallylargewithrespectto the manifoldonwhichitisacting,ourclassificationresultstatesthattheman- . h ifoldswhicharisearedescribedpreciselyasinvariantopensubsetsofcer- t a taincomplexflagmanifoldsassociatedtothecomplexifiedgroups. m [ 1 Introduction 1 v 0 In this paper we consider complex manifolds that admit “classical symme- 8 2 tries”,thatis,manifoldsXendowedwithalmosteffectiveactionsbyholomor- 4 phictransformationsofclassicalsimplerealLiegroupsG .HerewesaythatG 0 0 . 1 isaclassicalsimplegroupifitisconnectedanditsLiealgebrag0 isaclassical 0 simple realLie algebra, i.e. one of the following mutually exclusive possibil- 9 ities holds: (a) g is a real form of a classical simple complex Lie algebra; (b) 0 0 g isclassicalsimplecomplexLiealgebrag˜ regardedasarealalgebra(inthis : 0 0 v R casewewriteg = g˜ ). ThroughoutthepaperwerefertorealLiealgebrasof i 0 0 X these two kinds as algebras of types I and II, respectively. Let g be the com- r plexificationofg0. For g0 of typeI the algebragis aclassicalsimple complex a Liealgebra,andforg oftypeIIitisisomorphictog˜ ⊕g˜ . Wewillbeprimar- 0 0 0 ily interested in type I algebras. There are numerous examples of manifolds with classical symmetries in this case, perhapsthe best-known ones being ir- reducibleHermitiansymmetricspacescorrespondingtoclassicalLiealgebras (see[Hel],p. 354).MoregeneralexamplesaregivenbyopenG -invariantsub- 0 setsofcomplexflagmanifoldsG/P,whereGistheuniversalcomplexification ofG and Pisaparabolicsubgroupof G. HereG actsonG/Pbymeansofa 0 0 coveringmap G → Gˆ , where Gˆ ⊂ G isarealformof G. For resultsonthe 0 0 0 orbitstructureofactionsofrealformsoncomplexflagmanifoldswereferthe readerto[FHW],[Wo1],[Z],[ACM]andreferencestherein. Weareinterestedinobtaining,wheneverpossible,acompleteexplicitclassifi- cationof manifoldswithclassicalsymmetries foranychoiceof g . One moti- 0 2 AlanHuckleberryandAlexanderIsaev vationforourstudyisthefollowing“ballcharacterizationtheorem” obtained in[I1](cf.[BKS]). Theorem1.1. Let X be aconnected complexmanifoldofdimension n ≥ 2, and Bn theunitballinCn. AssumethatthegroupsAut (X)andAut (Bn)ofholomorphic O O automorphismsof X and Bn are isomorphic as topologicalgroups endowed with the compact-opentopology. Then X is biholomorphic to either Bn or to the complement of its closure in Pn. In particular, if in addition X is Stein or hyperbolic, then X is biholomorphictoBn. InthispaperwewillseethatTheorem1.1isacorollaryofgeneralclassification results for manifolds with classical symmetries. In fact, the conclusion of the theoremholdstrueifXadmitsanalmosteffectiveactionoftheclassicalsimple group PSU := SU /(center),whichisisomorphictoAut (Bn). Notethat n,1 n,1 O forG = PSU wehaveg =su andg=sl (C). 0 n,1 0 n,1 n+1 Let n := dimX (we assume that n ≥ 2), and define k(g ) for g of type I 0 0 (resp. typeII)tobethedimensionofthedefiningcomplexrepresentationofg (resp. g˜ ). Observethatifk(g ) ≤ n+1thereisnoreasonableclassificationof 0 0 manifolds X fortype I algebras(resp. type II algebras)if g (resp. g˜ )belongs 0 to the B- or D-series of simple complex Lie algebras (the B-series consists of so (C) with k = 2l+1, l ≥ 2, andthe D-seriesof so (C) with k = 2l, l ≥ 4). k k Indeed,anydirectproductQ ×Z,whereQ foranym ≥1denotesthe k(g0)−2 m m-dimensional projective quadric in Pm+1 and Z is anycomplex manifold of dimension n−k(g )+2, admits an almost effective action of the orthogonal 0 groupSO (C),andhenceactionsofallitsrealformsandtherealLiegroup k(g0) SO (C)R. Theactionisdefinedasthestandardactiononthefirstfactorand k(g0) thetrivialactiononthesecondfactor. For a similar reason, if k(g ) ≤ n one cannot explicitly classify manifolds X 0 for type I algebras(resp. type II algebras) if g (resp. g˜ ) belongs to the A- or 0 C-series(theA-seriesconsistsofsl (C)withk ≥ 2,andtheC-seriesofsp (C) k k with k = 2l, l ≥ 3). Indeed, any direct product Pk(g0)−1×Z, where Z is an arbitrary complex manifold of dimension n−k(g )+1, admits an almost 0 effectiveactionofthegroupsSL (C),Sp (C),againdefinedasthestan- k(g0) k(g0) dardactiononthefirstfactorandthetrivialactiononthesecondfactor. Even for n = 2,g = sl (R) anexplicitclassificationisunlikelytoexist. Indeed,in 0 2 [I5](seealso[I4])wedeterminedallhyperbolic2-dimensionalmanifoldsX,for whichthegroupAut (X)ofholomorphicautomorphismsofXhasdimension O 3.Thelistofsuchmanifoldsisquitelong,andoneofthemostdifficultpartsof the classification corresponds to the case when the Lie algebraof Aut (X) is O isomorphictosl (R)(cf. [I3]). Itispossiblethattheresultsof[I5]canbegen- 2 eralizedtothecasewhenagroupG withLiealgebrasl (R)actsonX almost 0 2 effectivelyandproperly,butitisunlikelythatthereexistsanexplicitclassifica- tioniftheassumptionofpropernessisdropped. Onthepositivesidewenote, however,thatin[IK2]allmanifoldsthatadmitaneffective actionofSU were n determined(cf.[U1]),andin[I2]weshowedthatanyhyperbolicmanifoldXof ClassicalSymmetriesofComplexManifolds 3 dimensionn ≥ 3withdimAut (X) = dimsl (C) = n2−1isbiholomorphic O n toaproductBn−1×S,whereSisaRiemannsurface. Followingtheabovediscussion,inthispaperwemakethefollowingassump- tion. We say that k(g ) and n satisfy Condition (>) for g of type I (resp. 0 0 type II), if k(g ) > n when g (resp. g˜ ) belongs to the A- or C-series, and 0 0 k(g ) > n+1 when g (resp. g˜ ) belongs to the B- or D-series. Note that the 0 0 algebra sp (C) is isomorphic to so (C) and we include it in the B-series, and 4 5 thealgebraso (C)isisomorphictosl (C)andweincludeitintheA-series. 6 4 It turns out that Condition (>) implies that k(g ) = n+1 for the case of the 0 A- and C-series and k(g ) = n+2 for the case of the B- and D-series. We 0 call these relations Condition (=). This condition implies, in particular, that themaximaldimensionofaclassicalsimple complexgroupthatcanacton X non-trivially is a polynomial in n. In contrast, it was shown in [SW] that for non-semisimple complex groupsthe dimension candepend exponentiallyon nevenifXiscompactandhomogeneous(butnotKa¨hler). WewillnowdescribethecontentofthepaperstartingwithalgebrasoftypeI. OurstudyofG -orbitsinX reliesonpassingtotheuniversalcomplexification 0 GofG (fortheconstructionofGsee[Ho]). ThegroupGactsonXlocally,but 0 this action may not be globalizable. We overcome this difficulty by means of fiberingeveryG -orbitY inX overaG -orbitYˆ inacomplexprojectivespace 0 0 Pℓ. Such a fibering ϕg0,Y : Y → Yˆ comes from the g0-anticanonical fibration associated to the orbit Y (we outline the construction and main properties of thisfibrationinSection2). SinceAut (Pℓ)isacomplexLiegroup,theuniver- O salitypropertyof G impliesthat G actsonPℓ globally andYˆ liesin a G-orbit Z = G/H. Moreover,themap ϕg0,Y extendstoaholomorphicmapψS defined on the union S of local G-orbitsof points inY, andthe set ψ (S) is open in Z S (seeSection2). Therefore,Zisatmostn-dimensional,thatis, H hascodimen- sionatmostninG(seeRemark2.1).ItthenfollowsthatHisapropermaximal parabolic subgroup of maximal dimension in G and thus Z is a complex flag manifold,onwhichG actsasarealformofG.TheparabolicityofHisaconse- 0 quenceofageneralresultondimension-theoreticallymaximalcomplexclosed subgroupsofcomplex Liegroupsthatwe obtaininProposition 3.1inSection 3. The proof of Proposition 3.1 utilizes Tits’ normalizer fibration for complex groups. Notethatforacomplexgroupthecorrespondinganticanonicalfibra- tioncoincideswiththenormalizerfibration(see[HO],p. 65). Infact, Z isbiholomorphic toPn ifgbelongs tothe A-orC-series, andtothe n-dimensionalprojectivequadric Q if gbelongs tothe B-or D-series, except n for the case of so (C), where both P3 and Q occur. Under the equivalence 5 3 Z ≃PnthegroupGactsonPnaseitherPSL (C)orPSp (C)(embedded n+1 n+1 inPSL (C)inthestandardway),andundertheequivalenceZ ≃ Q itacts n+1 n onQ as PSO (C). Further, themapψ extendstoalocallybiholomorphic n n+2 S G -equivariant surjective map ψ : X → U, where U is an open G -invariant 0 0 subsetof Z. In Sections 6–8 we go over allclassicalsimple complex Lie alge- 4 AlanHuckleberryandAlexanderIsaev brasgandtheirrealformsg andshowthatonalltheoccasionsthemapψisin 0 factabiholomorphism. Thus,ourclassificationofmanifoldsX fortypeIalge- brasg consistsofallopenconnected G -invariantsubsetsofeitherPn orQ . 0 0 n The determination of all open G -invariant subsets is an interesting question 0 initsownright,especiallyfortheactionsoftherealformsof PSO (C)onQ , 8 6 where the triality property plays a role. To establish that ψ is biholomorphic we analyze the specific orbit structure for the G -actionon Z in eachcase, so 0 ourproofofthebiholomorphicityofψisacase-by-caseargumentinterwoven withthedeterminationofopenG -invariantsubsetsofZ. 0 We now summarize the content of the precedingthree paragraphsin the fol- lowingtheorem,whichisthemainresultofthepaper. Theorem1.2. LetXbeaconnectedcomplexmanifoldofdimensionn ≥2admitting an almost effective action by holomorphic transformations of a connected simple Lie group G . Let g be theLie algebra of G , and assume that g is of typeI. Suppose, 0 0 0 0 furthermore,thatk(g )andnsatisfyCondition(>). Thenk(g )and nsatisfyCon- 0 0 dition(=),andXisbiholomorphictoaconnectedG -invariantopensubsetoftheflag 0 manifoldZ = G/H,whereGistheuniversalcomplexificationofG ,thesubgroupH 0 is a maximalparabolic subgroup of maximaldimensionin G, and G actson Z as a 0 realformofG. Theorems6.2,6.3,6.4giveadetailedclassificationofmanifoldsX forthecase oftheA-series,Theorems7.2,7.4forthecaseoftheC-series,Theorems8.2,8.3, 8.5,8.6,8.7,8.8forthecaseoftheB-andD-series. The above discussion for type I algebras holds for type II algebras as well, with the differencethat G actson Z aseither PSL (C)R or PSp (C)R if 0 n+1 n+1 Z is biholomorphic to Pn, and as PSO (C)R if Z is biholomorphic to Q . n+2 n All these actions are transitive on Z. This makes the case of type II algebras significantlyeasierthanthatoftypeIalgebrasandleadstothefollowingresult. Theorem 1.3. Let X be a connected complex manifold of dimension n ≥ 2 admit- tingan almosteffectiveactionby holomorphictransformationsof a connectedsimple Lie group G . Let g be theLie algebra of G , and assume thatg is oftypeII. Sup- 0 0 0 0 pose,furthermore,thatk(g ) andnsatisfyCondition(>). Thenk(g )andnsatisfy 0 0 Condition(=), and X isbiholomorphictoa flagmanifold Z = G/H, where G isthe universalcomplexificationof G ,and H isamaximalparabolicsubgroupofmaximal 0 dimensioninG. Theorem1.3isprovedinSection4,whereadetailedversionofthistheoremis alsostated(seeTheorem4.1). It should be stressed here that we utilize the g -anticanonical fibration in the 0 proofsofTheorems1.2,1.3becausethelocalactionofGonX isnotknownto beglobalizableapriori. IfG actedon X globally,thenusingtheanticanonical fibration would only be required in the proof of Proposition 3.1 for complex groupsactingtransitively. ClassicalSymmetriesofComplexManifolds 5 Beforeproceedingwenote thatsmooth actionsof variousclassicalgroupson specialrealmanifoldsofspecificdimensions(thatareoftenassumedtobeodd) havebeenextensivelystudied. Wereferthereader,forinstance,to[U1]–[U3], [UK],[T]andreferencesthereinfordetails. Acknowledgements. Thisworkwasinitiatedwhilethefirstauthorwasvisit- ingtheAustralianNationalUniversity. TheresearchissupportedbytheAus- tralianResearchCouncil. 2 g -Anticanonical Fibration 0 Inthissectionweintroduceatoolthatwillbefrequentlyusedthroughoutthe paper.Moredetailsonthissubjectcanbefoundin[HO],[Huck]. Let X be a connected complex manifold and G a connected real Lie group 0 acting on X almost effectively by holomorphic transformations. Denote by K−1 the anti-canonical line bundle over X, and let Γ(X,K−1) be the complex vector space of holomorphic sections of K−1. Clearly, G acts on Γ(X,K−1) 0 by complex-linear transformations. Let V be a complex G -stable finite- 0 dimensionalsubspaceofΓ(X,K−1). Considerthemeromorphicmap ϕ : X →P(V∗), ϕ (x):=[f ], with f (σ):=σ(x), x ∈ X, σ ∈V, V V x x (2.1) whereP(V∗)istheprojectivizationofthedualspaceV∗ ofV,and[a] ∈P(V∗) denotestheequivalenceclassofa ∈ V∗\{0}. Theindeterminacysetof ϕ co- V incideswiththesetofpointsxforwhich f ≡0.Notethatintheaboveformula x thevalue σ(x),if itisnon-zero, isonly well-definedfor aparticularchoice of coordinatesnearxandismultipliedbyaconstantindependentofσwhenone passestoanothercoordinatesystem(thisexplainstheneedtoconsiderP(V∗) instead of V∗ in the definition of ϕ ). Next, the action of G on V induces an V 0 actionofG onV∗ bycomplex-lineartransformations,andhenceanactionon 0 P(V∗). Themap ϕ isG -equivariantwithrespecttothisinducedaction. V 0 Onecanalsodefine ϕ byassigningtoeverypoint x ∈ X thesetH := {σ ∈ V x V : σ(x)=0},wherethehypersurfaceH isregardedasanelementofP(V∗). x AssumenowthatG actsonXtransitively. Inthiscaseonecanmakeapartic- 0 ularchoiceofV forwhich ϕ isholomorphiconallofX. LetΓ (X,TX)bethe V O spaceofvectorfieldsonX whichareholomorphic,i.e.,Z ∈ Γ (X,TX)ifand O onlyif C 1 Z := (Z−iJZ) 2 is a holomorphic (1,0)-fieldon X, where J is the operator of almost complex structure on X. Denote by g the Lie algebra of G and for v ∈ g let X ∈ 0 0 0 v Γ (X,TX)bethecorrespondingcompleteholomorphicvectorfieldonX,that O is, d X (x):= [exp(−tv)x]| , x ∈ X. v t=0 dt 6 AlanHuckleberryandAlexanderIsaev Since the action of G on X is almost effective, the map ι : v 7→ X is a 0 v Lie algebra isomorphism onto its image. Next, consider the Lie subalgebra gˆ := ι(g )+Jι(g ) ⊂Γ (X,TX).Clearly,gˆ isacomplexLiealgebrawiththe 0 0 0 O 0 C operatorofcomplexstructureinducedby J. BythecomplexificationZ 7→ Z C C werealizegˆ asthealgebraofholomorphic(1,0)-fieldsoftheformX +iX 0 v1 v2 forv ,v ∈g . 1 2 0 Let n := dimX. Set Vg0 := ngˆ0. Clearly, Vg0 is a finite-dimensional linear subspace of Γ(X,K−1) and isVspanned by sections of Γ(X,K−1) of the form σ = Z ∧···∧Z ,Z ∈ gˆ . Foreverysuchσandeveryg ∈ G theactionof g 1 n j 0 0 C C onσisgivenbygσ =(gZ )∧···∧(gZ ),whereforZ =X +iX wehave 1 n v1 v2 C C gZ :=X +iX . Adg(v1) Adg(v2) Since gZ liesingˆ0,itfollowsthatVg0 isG0-stable. Therefore,wecanconsider the corresponding map ϕg0 := ϕVg0 (see (2.1)). Since for every x ∈ X there existsσ = Z1∧···∧Znwithσ(x)6=0,themap ϕg0 isholomorphiconX. Incoordinatesthemap ϕg anditsequivariancepropertycanbedescribedas 0 follows. We let m+1 be the dimension of Vg0, and fix a basis σ0,...,σm in Vg0. Then for x ∈ X wehave ϕg0(x) = [σ0(x) : ··· : σm(x)]inhomogeneous coordinatesinPm. Further,for g ∈ G let A ∈ GL (C)bethelineartrans- 0 g m+1 formation by which g acts on Vg , written in coordinates with respect to the 0 basisσ0,...,σm. IdentifyingVg0 andCm+1 bymeansof thesecoordinates, we define an action of G on Cm+1 as g 7→ (A−1)T. This action induces an ac- 0 g tion on Pm, and it is straightforward to verify that ϕg (gx) = gϕg (x) for all 0 0 x ∈ Xandg∈ G0,i.e.,ϕg0 isG0-equivariant. Therefore,ϕg0(X)isaG0-orbitin Pm. Let H and J bethe isotropysubgroupsof apoint x ∈ X andthepoint 0 0 0 ϕg0(x0) ∈ Pm, respectively. Clearly, J0 contains H0, and the corresponding fibration G /H → G /J , 0 0 0 0 iscalledtheg -anticanonicalfibration. OnecanshowthatJ liesinthenormal- 0 0 izer N (H◦)oftheidentitycomponent H◦ of H in G (see[HO],pp. 61-62). G0 0 0 0 0 Animportantpropertyoftheg -anticanonicalfibrationisthatitisaholomor- 0 phicfiberbundlewithfiber(J /H◦)/(H /H◦),where J /H◦ isacomplexLie 0 0 0 0 0 0 group(see[HO],p. 64). WenowdroptheassumptionofthetransitivityoftheG -actiononX. LetY ⊂ 0 XbetheG -orbitofapointx ∈ X. Foreveryy ∈YthetangentspaceT (Y)to 0 0 y Y atyisspannedbythevectorsX (y)with v ∈ g . LetTˆ (Y)bethecomplex v 0 y subspace in T (X) spanned by the values of vector fields from gˆ = ι(g )+ y 0 0 Jι(g ) atthe point y. Since G actson X byholomorphic transformations, the 0 0 dimensionofthemaximalcomplexsubspaceofT (Y)isindependentofy∈Y. y HencethedimensionofTˆ (Y)isindependentofyaswell,andwedenoteitby y µ. Supposethatµ ≥ 1. Letg := g ⊕ig bethecomplexificationofg , andletG 0 0 0 beanyconnectedcomplexLiegroupwithLiealgebrag. Thealgebragactson ClassicalSymmetriesofComplexManifolds 7 X via the homomorphism τ : g → gˆ , τ(v +iv ) := X +JX . The map τ 0 1 2 v1 v2 inducesalocalholomorphic actionofG on X,andfory ∈ Y wedenotebyO y the local G-orbitof y. Clearly,O isa complex µ-dimensional submanifold of y X, and Tˆ (Y) is thetangentspacetoO aty. LetS := ∪ O . The setS isa y y y∈Y y (possibly non-injectively) immersed complex submanifold of X of dimension µcontainingY. Asbefore,werealizegˆ asthefinite-dimensionalcomplexLiealgebraofholo- 0 morphic(1,0)-fieldsonXoftheformXC+iXCforv ,v ∈g .SetV := µgˆ . v1 v2 1 2 0 S 0 Clearly,V isafinite-dimensionallinearsubspaceofΓ(X,K−1)andisspaVnned S bysectionsofΓ(X,K−1)oftheformZ ∧···∧Z ,Z ∈gˆ . Weletℓ+1bethe 1 µ j 0 dimensionofVS andfixabasisσ0,...,σℓinVS. Nowforx ∈ Sweset ψS(x) :=[σ0(x): ···: σℓ(x)]∈Pℓ. Clearly,ψ iswell-definedandholomorphic onS. Asbefore,the G -actionon S 0 V induces an action of G on Cℓ+1 by complex-linear transformations, and S 0 hence an action on Pℓ. It is straightforward to verify that ψ (gy) = gψ (y) S S for all y ∈ Y and g ∈ G0, that is, the map ϕg0,Y := ψS|Y is G0-equivariant. Furthermore, the map ψ is equivariantwith respectto the local G -actionon S 0 sSu.bIgtrfooullposwosftthhaetpϕogi0n,Yt(sYx)0iasnadGp00-o:r=bitϕign0,YP(xℓ.0)L∈etPHℓ0,arensdpeJ0ctbiveetlyh.eWisoethroapvye H ⊂ J ,andthecorrespondingfibration 0 0 G /H → G /J , 0 0 0 0 iscalledtheg -anticanonicalfibrationassociatedtotheorbitY. Arguingasin 0 theproofofProposition1onpp. 61-62of[HO],weseethat J liesinN (H◦). 0 G0 0 We now assume thatthe complexification g of g is semisimple, and let G := 0 C G betheuniversalcomplexificationofG (see[Ho]).ThegroupG ismapped 0 0 0 intoGbymeansofahomomorphismγsuchthatγ(G )isaclosedrealformof 0 G. SincethecenterofanyconnectedcomplexsemisimpleLiegroupisfinite,it followsthatγislocallyinjective,hencetheLiealgebraofGisisomorphictog. Further,aswehaveseen,theactionofG onV inducesanactionofG onPℓ, 0 S 0 thatis,ahomomorphism ρ : G0 → AutO(Pℓ) ≃ PSLℓ+1(C). Since PSLℓ+1(C) isacomplexLiegroup,theuniversalitypropertyofGimpliesthatthereexists acomplexLiegrouphomomorphismρC : G → PSLℓ+1(C)suchthat C ρ= ρ ◦γ. (2.2) In fact, since ρ = π◦ρ˜, where ρ˜ : G0 → SLℓ+1(C) is a linear representation andπ : SLℓ+1(C)→ PSLℓ+1(C)isthenaturalfactorizationmap,thereexistsa linearrepresentationρ˜C : G → SLℓ+1(C)suchthatρ˜ = ρ˜C◦γandρC =π◦ρ˜C. Thus,thegroupGholomorphicallyactsonPℓ bywayofρC,andwehavethe inclusionoforbitsG .p ⊂ G.p . NotethatifGisasimplegroup,eitheritacts 0 0 0 ontheorbitG.p almosteffectively,or G.p = {p }. Inthelattercase J = G , 0 0 0 0 0 8 AlanHuckleberryandAlexanderIsaev hence H◦ isnormalinG . SinceG issimple,itfollowsthateither H = G ,or 0 0 0 0 0 H isdiscrete. 0 C Let γ : U → G be a continuation of γ to a local complex Lie group homo- C morphism defined in a neighborhood U of e ∈ G such that d γ (v +iv ) = e 1 2 d γ(v )+Jd γ(v )forallv ,v ∈g ,whered denotesthedifferentialateand e 1 e 2 1 1 0 e C J istheoperatorofcomplexstructureintheLiealgebraofG.Clearly,γ maps U ontoaneighborhoodoftheidentityinG. ThenthelocalG -equivarianceof 0 ψ andproperty(2.2)implythatforeveryx ∈Swehave S C ψ (gx)=γ (g)ψ (x), (2.3) S S where g ∈ G issufficientlyclosetoe. Inparticular,ψ (S)isanopensubsetof S theorbitG.p . SincedimS= µ≤ n,itfollowsthatdimG.p ≤ n. 0 0 Remark2.1. LetH betheisotropysubgroupof p withrespecttotheG-action. 0 SincedimG.p ≤ n,thecodimensionof H in G isatmostn. Thisobservation 0 will be important for our futureapplications. In fact, in allsituations consid- ered below H will turn out to be a proper dimension-theoretically maximal subgroupofG. Suchsubgroupsarestudiedinthenextsection. Clearly, the above construction of the g -anticanonical fibration requires the 0 dimension µ of S to be positive and does not apply in the case when x is a 0 fixedpointofthe G -actiononX. Wewillnowstateasimplelemmathatwill 0 beusefulforrulingoutfixedpointslaterinthepaper. Lemma2.2. Let G beaconnectedsimplerealLiegroupactingalmosteffectivelyby 0 holomorphictransformationsonann-dimensionalcomplexmanifoldX. Assumethat G fixes a point in X and has a positive-dimensional compact subgroup. Then the 0 complexificationgoftheLiealgebrag ofG hasanon-trivialcomplexn-dimensional 0 0 representation. Proof. Letx beafixedpointoftheG -action. Considertheisotropyrepresen- 0 0 tationofG atx 0 0 α : G → GL(T (X),C), g 7→d g, x0 0 x0 x0 where GL(T (X),C) ≃ GL (C) is the group of non-degenerate complex- x0 n lineartransformationsofthetangentspaceT (X),andd gdenotesthediffer- x0 x0 entialatx oftheholomorphicautomorphismbywhichanelementg ∈ G acts 0 0 onX. LetGbetheuniversalcomplexificationofG . SinceGissemisimple,its 0 Liealgebracoincideswithg. TheuniversalitypropertyofGimpliesthatthere existsalinearrepresentationαC : G → GL (C)suchthatα = αC ◦γ,where x0 n x0 x0 γ : G → G is a locally injective homomorphism such that γ(G ) is a closed 0 0 realformofG. LetK beamaximalcompactsubgroupofG . TheactionofK onX isalmost 0 0 0 effectiveand canbe linearized near x , which implies that the representation 0 α| is almost faithful. Since K is positive-dimensional and G is simple, it K0 0 0 ClassicalSymmetriesofComplexManifolds 9 C followsthatαisalmostfaithful,andthereforeα | isalmostfaithful.Since x0 γ(G0) C γ(G ) ispositive-dimensional, weseethatkerα 6= G,andthusghasanon- 0 x0 trivialcomplexn-dimensionalrepresentationasrequired. Remark 2.3. Observe that in Lemma 2.2 the group G is simple if g is of type 0 I, and G is a locally direct product G = G ·G , where each G is a closed 1 2 j subgroupofGwithLiealgebrag˜ ,ifg isoftypeII.Henceforg oftypeIthe 0 0 0 C representationα is almost faithful and thus the induced representationof g x0 isfaithful. Ifg isoftypeIIandthekerneloftherepresentationofg = g˜ ⊕g˜ 0 0 0 is non-trivial, then this kernel coincides with one of the simple factors. We alsoremarkthatthestatementofLemma2.2holdstrueifG isnotnecessarily 0 semisimple, but is assumed instead to have no positive-dimensional normal closedsubgroupLsuchthattheintersectionL∩K isdiscrete. Inthiscaseαis 0 almostfaithful,andtheproofgivenaboveapplies. 3 Dimension-Theoretically Maximal Subgroups of Complex Lie Groups Our arguments in the forthcoming sections rely on the fact that a proper dimension-theoretically maximal subgroup of a connected complex semisim- ple Lie group is parabolic. This result is obtained in the present section. In fact,weprovethe following moregeneralproposition, which–atleastinthe semisimplecase–iswell-knowntospecialists(cf. [Wi],p. 46,Lemma1). Proposition3.1. LetGbeacomplexconnectedLiegroupandH ⊂ Gaproperclosed complexsubgroup. Assumethat H isdimension-theoreticallymaximal,thatis, there existsnoproperclosedcomplexsubgroupofGofdimensiongreaterthandimH. Then oneofthefollowingholds: (i)G′ ⊂ H,andtheAbeliangroupG/Heitherhasdimension1,orisacompacttorus withoutnon-trivialpropersubtori; (ii)G/HisbiholomorphictoCp for p≥1; (iii)HcontainstheradicalRofGandH/RisamaximalparabolicsubgroupofG/R ofmaximaldimension. If, furthermore, G is semisimple, then H is a maximal parabolic subgroup of G of maximaldimension. Proof. Let N := N (H◦)bethenormalizeroftheidentitycomponent H◦ of H G in G. Since H is dimension-theoretically maximal, we have either N = G, or N◦ = H◦. Wewillconsiderthesetwocasesseparately. Case1. Supposefirstthat N = GandletG := G/H◦. Since H isdimension- 1 theoreticallymaximal, G hasnopositive-dimensionalproperclosedcomplex 1 10 AlanHuckleberryandAlexanderIsaev subgroups. LetG = R ·S betheLevi-MalcevdecompositionofG,whereR 1 1 1 1 is the radicaland S is a semisimple Levisubgroup of G , respectively. Since 1 1 R isclosedandconnected,iteithercoincideswithG oristrivial. Inthelatter 1 1 case G is semisimple and hence contains a closed copy of SL (C), which in 1 2 turncontains aclosed copyof C∗. Thiscontradiction impliesthat G = R is 1 1 solvable.SinceG issolvable,thereexistsapositive-dimensional(notnecessar- 1 ily closed) complex Abeliansubgroup A in G . Let C (A) be the centralizer 1 G1 of A in G . Since C (A) is a closed complex subgroup of G containing A, 1 G1 1 we have C (A) = G . Hence A is contained in the center Z(G ) of G , and G1 1 1 1 thereforeZ(G )ispositive-dimensional. SinceZ(G )isaclosedcomplexsub- 1 1 groupofG ,wehaveZ(G ) = G ,thatis, G isAbelian. Therefore,weobtain 1 1 1 1 that G′ ⊂ H◦, andhence G := G/H isacomplexAbeliangroup thathasno 2 positive-dimensionalproperclosedcomplexsubgroups. By Theorem 3.2 of [Morim], any complex Abelian group is isomorphic to a direct product Q×Cm×(C∗)k, where Q is a complex Cousin group (i.e. Q doesnotadmitanynon-constantholomorphic functions). Itthenfollowsthat G iseitheraCousingroupor1-dimensional. 2 Lemma3.2. ACousingroupthathasnopositive-dimensionalproperclosedcomplex subgroupsisisomorphictoacompacttorus. Proof. Let Q bea Cousin group satisfying the assumptions of the lemma. We write it as Q = Cℓ/Γ for some ℓ, where Γ is a non-trivial discrete subgroup in Cℓ. Let V be the real subspace of Cℓ spanned by Γ. Then K := V/Γ is the maximal compact subgroup of Q. Assuming that Q is non-compact, we see that V is a positive-dimensional proper real subspace of Cℓ. Since Q has nopositive-dimensionalproperclosedcomplexsubgroups,Visnotacomplex subspace. LetW be a complement to V∩iV in V, such that the realAbelian group W/W ∩Γ is compact (and hence is isomorphic to (S1)s for some s > 0). Then (W+iW)/W ∩Γ is a complex closed subgroup of Q isomorphic to (C∗)s. SinceQhasnopositive-dimensionalproperclosedcomplexsubgroups, itfollowsthatQisisomorphicto(C∗)s,whichcontradictstheassumptionthat QisaCousingroup. Thus,wehaveshownthatQiscompactasrequired. Lemma 3.2 yields that the group G either has dimension 1, or is a compact 2 toruswithoutnon-trivialpropersubtori. Thisisoption(i)oftheproposition. Case 2. Assume now that N◦ = H◦ and consider the normalizer fibration G/H → G/N with finite fiber N/H. This fibration coincides with the g- anticanonical fibration, where g is the Lie algebra of G (see [HO], p. 65). In particular, there is a homomorphism ρ : G → PSL (C) = Aut (Pr) such r+1 O thatN =ρ−1(N),whereN consistsofallelementsofρ(G)thatfixsomepoint x inPr. Thehomomorphismρisthecompositionofalinearrepresentationof 0 GinGL (C)andthenaturalfactorizationmapGL (C) → PSL (C). By r+1 r+1 r+1 aresultduetoChevalley[C](seealso[HO],p. 31),thecommutatorsubgroup ρ(G)′ = ρ(G′)isanalgebraicsubgroupofPSL (C).Hencetheorbitρ(G)′.x r+1 0 isclosed in ρ(G).x , andthusthe subgroup P := ρ−1(ρ(G)′)N isclosed in G. 0