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Advances in Mathematics
www.elsevier.com/locate/aim
Frankel conjecture and Sasaki geometry
Weiyong Hea,1,Song Sunb,c,∗,2
a DepartmentofMathematics,UniversityofOregon,Eugene,OR,97403,USA
b DepartmentofMathematics,ImperialCollege,LondonSW72AZ,UK
cDepartmentofMathematics,SUNY,StonyBrook,NY11794,USA3
a r t i c l e i n f o a bs t r a c t
Article history:
Received24September2015 Accepted29November2015 Availableonline4February2016 CommunicatedbyTomaszS.
Mrowka
Keywords:
Sasakimanifolds Frankelconjecture Positivity
We classify simply connected compact Sasaki manifolds of dimension 2n+ 1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to Cn+1\{0}. As an application we recover the theorem of Mori and Siu–Yau on theFrankelconjecture andextendit to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standardround sphere in twosteps, both fixing the complex structure on the Kähler cone. First, we deform the metric along the Sasaki–Ricci flow and obtain a limit Sasaki–Ricci solitonwith positive transverse bisectionalcurvature.ThenbyvaryingtheReebvectorfield whichessentiallydecreasesthevolumefunctional,wedeform the Sasaki–Ricci soliton to a Sasaki–Einstein metric with positivetransversebisectionalcurvature,i.e.aroundsphere.
The second deformation is only possible when one treats simultaneously regular andirregular Sasaki manifolds,even ifthemanifoldonestartswithisregular(quasi-regular),i.e.
Kählermanifolds(orbifolds).
© 2016ElsevierInc.All rights reserved.
* Correspondingauthor.
E-mailaddresses:whe@uoregon.edu(W. He),s.sun@imperial.ac.uk,song.sun@stonybrook.edu (S. Sun).
1 PartiallysupportedbyaNationalScienceFoundation grant,awardNo.DMS-1005392.
2 PartiallysupportedbyEuropeanResearchCouncilawardNo. 247331.
3 Newaddress.
http://dx.doi.org/10.1016/j.aim.2015.11.053 0001-8708/© 2016ElsevierInc.All rights reserved.
1. Introductionandmain results
Inthispaperwestudy compactSasaki manifoldswithpositivetransversebisectional curvature. Sasakigeometry, inparticular, Sasaki–Einstein manifoldshavebeen studied extensively. Readers are referred to the monograph [5], the survey paper [50] and the referencesthereinforthehistoryandrecentprogressonthis subject.
Thestudyofmanifoldswithpositivecurvatureisoneofthemostimportantsubjects inRiemanniangeometry.There isalot ofrecentdeepprogressonthis,especiallyusing the techniqueof Ricciflow, see [4] and [9] for example. In Kähler geometry anatural conceptisthepositivityofthebisectionalcurvature. ItwasconjecturedbyFrankel[21]
thatacompactKählermanifoldofcomplexdimensionnwithpositivebisectionalcurva- tureisbiholomorphic tothecomplexprojective spaceCPn.TheFrankelconjecturewas provedinlater1970sindependentlybyMori[42](heprovedthemoregeneralHartshorne conjecture)viaalgebraicgeometryandSiu–Yau[48]viadifferentialgeometry.Sasakige- ometry is an odddimensional companion of Kähler geometry, so it is very naturalto ask forthe counterpartof thetheorem of Mori andSiu–Yau onthe Frankelconjecture forSasaki manifolds. This isthe major point ofstudy inthis article. Wewouldlike to emphasizethatthis generalizationseemsto beinteresting in thatitprovides auniform frameworkwhichalsoprovestheoriginalFrankelconjecture,bydeformationtocanoni- calmetrics,as attemptedpreviouslybymanypeople(cf. [15,16,44]).Moreover, theuse ofSasaki geometry also yieldscertain orbifoldversionof theFrankelconjecture, which seemsto be difficult to obtainwith the knownapproaches. Finally, as already pointed outin [4] apinching towardsconstant curvature proof of the Frankelconjecture using Ricci flow seems notplausible, as there are examples of two dimensional Riccisoliton orbifoldswithpositivecurvature.Oneoftheapplicationsoftheresultsdevelopedinthis articleistoclassifysuchsolitons,inauniform way.
Sasaki geometry indimension 2n+ 1 is closely related to Kähler geometry inboth dimensions2(n+ 1) and2n.A Sasaki manifoldM ofdimension2n+ 1 admits, onone hand, aKähler cone structure on the product X = M×R+, and on the other hand, a transverseKählerstructureonthe(local)quotientbytheReebvectorfield.Fornowwe viewaSasakistructure onM asaKählerconestructure onX,andweidentifyM with thelink{r= 1}inX.A standardexampleofaSasaki manifoldistheodddimensional roundsphereS2n+1. Thecorresponding KählerconeisCn+1\{0}with theflatmetric.
A Sasaki manifold admits a canonical Killing vector field ξ, called the Reeb vector field.Itisgivenbyrotatingthehomotheticvectorfieldr∂ronX bythecomplexstruc- tureJ.The integralcurvesof ξare geodesics,and giveriseto afoliationonM,called theReeb foliation.Then there isaKählerstructure onthelocal leafspace oftheReeb foliation, called thetransverse Kähler structure. Ifthe transverse Kähler structure has positivebisectionalcurvature, wesaytheSasaki manifoldhaspositivetransverse bisec- tionalcurvature.IftheSasakimanifoldhaspositivesectionalcurvature,itautomatically haspositive transverse bisectional curvature, for example,the round metric onS2n+1.
Indeedfor theroundsphereS2n+1, thetransverseKähler structureontheleafspace is isometric totheFubini–StudymetriconCPn.
The main goal of this article is to classify compact Sasaki manifolds with positive transversebisectionalcurvature.Byahomothetictransformationsuchmanifoldsalways admit Riemannian metricswith positiveRiccicurvature, so theymust havefinite fun- damental group.Thereforewithoutlossofgenerality,wemayassumethemanifoldsare simplyconnected. Ourmainresultis
Theorem 1.1. Let (M,g) be a compact simply connected Sasaki manifold of dimension 2n+1withpositivetransversebisectionalcurvature,thenitsKählercone(X,J)isbiholo- morphictoCn+1\{0}.Moreover,M isaweightedSasakisphere,i.e.M isdiffeomorphic to thesphere S2n+1 and theSasaki metric isasimpleSasakimetricon S2n+1.
Roughly speaking, a simple Sasaki metric on S2n+1 is aSasaki metric thatcan be deformed to the round metric on S2n+1 through a simple deformation. The relevant definitions will be given in Section2. When n = 1, our proof implies that any Sasaki structureonS3issimpleanditsKählerconeisC2\{0};thisresultwasprovedbyBelgun [2]as apartoftheclassificationofthreedimensionalSasakimanifolds.Inasequel[33], wewillusetheresultsofthispapertogetherwiththetechniqueofBrendle–Schoen[8](cf.
also [29])to classifycompactSasaki manifoldswithnon-negative transversebisectional curvature.
As adirectconsequenceof Theorem 1.1,we obtaintheclassificationof compactpo- larized orbifolds withpositivebisectionalcurvature.
Corollary1.1.Acompactpolarizedorbifold(M,J,g,L)withpositivebisectionalcurvature is bi-holomorphictoafinite quotientof aweightedprojective space.
Thenotionpolarizedorbifold istakenfrom[45].Bythiswemeanthereisanorbi-line bundle L, and inany orbifoldchart (Up,Lp,Gp) the actionof Gp on Lp is faithful.As a special case of Corollary 1.1, we obtain an alternative analytic proof of Siu–Yau’s Theorem.
Corollary 1.2.(See[42,48].)A compactKähler manifoldwithpositivebisectionalcurva- tureisbi-holomorphic tothecomplexprojective space.
Oneinterestingpoint hereisthatourproofofCorollaries 1.1 and1.2 dorelyon the frameworkofSasaki geometry.A converseofTheorem 1.1 isalsotrue.
Theorem 1.2. Any simple Sasaki structureon S2n+1 can bedeformed toa Sasaki–Ricci solitonwithpositivetransversebisectionalcurvature,byatransverseKählerdeformation.
In particular, a weighted projective space carries an orbifold Kähler–Ricci soliton with positive bisectionalcurvature.
The existence of Sasaki–Ricci solitons on weighted Sasaki sphere follows from the result of Futaki–Ono–Wang [23] on toric Sasaki manifolds. We will prove that these Sasaki–Riccisolitons allhavepositivetransversebisectional curvature.Weremarkthat herethesemetricsarenotexplicit,andwearenotabletofindageneralwayofproducing anexplicit orbifoldKählermetricwith positivebisectional curvatureeven onweighted projectivespaces.
BeforewesketchthemainideastoproveTheorem 1.1,itisvaluabletorecalltheknown proofs of the Frankelconjecture. In both [42] and [48] theexistence of rational curves plays anessentialrole. Mori provedamoregeneralresultthataFanomanifold always containsarationalcurvebyabend-and-breakargument andthealgebraic geometryin positivecharacteristic;whileSiu–YauusedtheSacks–Uhlenbeckargumenttoproducea stableharmonicsphere,andexploitedthepositivityofbisectionalcurvaturetoprovethat suchasphereiseitherholomorphicoranti-holomorphic.A keyingredientintheproofof Siu–YauisacharacterizationofthecomplexprojectivespacebyKobayashi–Ochiai[37].
There is, to the authors’ knowledge so far, no analogue of this in Sasaki geometry to characterizeaweightedSasaki sphere,orparticularly aweightedprojectivespace. This seemsto beamajorobstacleforadaptingtheapproachofSiu–YautotheSasaki case.
We proceed along adifferent track inthis paper, thatis, by deforminga geometric structurenaturallytoastandardonethatcanbeclassifiedmore.Inearly1980sHamilton [30]introducedtheRicciflow,asapowerfultooltoevolveRiemannianmetricstowards canonicalmodels.OnKählermanifoldstheRicciflowpreservestheKählercondition.It iscalledtheKähler–Ricciflow andwasfirststudiedbyCao[12].Bando[1](forcomplex dimension three) and Mok [41] (for all dimensions) studied the Kähler–Ricci flow on compactmanifoldswithpositive(non-negative) bisectionalcurvature.Theyprovedthat thispositivity(non-negativity) ispreservedalong theflow, usingHamilton’s maximum principlefortensors. Sincethen, therehavebeen manyattempts toseek aproof ofthe Frankelconjectureusing Ricciflow,and therehasbeen extensivestudy ofKähler–Ricci flowwithpositive(nonnegative)bisectionalcurvature.Wemention[15,16,13,44]toname afew.NotethatBerger[3]provedthataKähler–Einsteinmetricwithpositivesectional curvatureisisometrictothecomplexprojectivespacewiththeFubini–Studymetric;this resultwaslatergeneralizedtoKähler–Einsteinmanifoldswithpositivebisectionalcurva- turebyGoldberg–Kobayashi[25]wheretheyfirstintroducedtheconceptofholomorphic bisectionalcurvature.OnecangetanalternativeproofoftheMori’sandSiu–Yau’sthe- oremontheFrankelconjectureiftheKähler–Ricciflowconverges toaKähler–Einstein metricwithpositivebisectionalcurvature.
LateronPerelmanintroducedmanyrevolutionaryideas,includingby nowwell-known entropyfunctionals[43]into thestudyoftheRicciflow,whichleadhimtothesolution of the Poincaré conjecture and Thurston’s geometrization conjecture. He also proved verydeep resultsfor theKähler–Ricci flow on Fanomanifolds, namely, thatthescalar curvatureand thediameterare uniformlybounded alongtheflow; detailsof hisresults can be found in [47]. Combining this with Mok’s results, it then easily follows that theKähler–Ricciflow convergesbysequence toaKähler–Ricci solitonupto diffeomor-
phisms, ifthe initialmetrichaspositive(nonnegative) bisectionalcurvature. Using the Morse–Bott theory and dimension induction, Chen, Tian and the second author [14]
gave a direct proof that the limit Kähler–Ricci soliton, and hence the original Kähler manifold is biholomorphic to the complexprojective space. The proof still depends on producingrationalcurvesandapplyingtheresultsofKobayashi–Ochiai.AlongtheRicci flow weonlyknow thataKähler–Ricci soliton withpositivebisectionalcurvaturemust be Kähler–Einsteinaposteriori, andadirectproofofthisisstill lacking.
Given the analogue between them, many concepts, techniques and results can be carriedoverfromKählergeometrytoSasakigeometrywithcertainmodifications.Sasaki–
RicciflowwasintroducedbySmoczyk–Wang–Zhang[49]asacounterpartofKähler–Ricci flow;itdeformsaSasakimetricsuchthatitstransverseKählermetricisdeformedbythe transverse Kähler–Ricci flow. Independently, Collins [18] andthe first author [31] gen- eralizedPerelman’sentropyandcorrespondingresultsintheKähler–RicciflowonFano manifolds to theSasaki settingbyconsideringonly basic geometricdata, i.e.geometric quantitiesthatareinvariantalongtheReebvectorfields.ThentheW functionalismono- tonealongSasaki–Ricciflow,andthe(transverse)scalarcurvatureandthediameterare bothuniformlyboundedalongtheflow.Furthermore,thefirstauthorstudiedtheSasaki–
Ricciflowwithpositive(nonnegative)transversebisectionalcurvature.It isshownthat theflowconvergestoaSasaki–Riccisolitonwithpositivetransversebisectionalcurvature.
Itisalsoprovedin[31]thatacompact,simplyconnectedSasaki–Einsteinmanifoldwith positive transversebisectional curvature hasconstant transverseholomorphic sectional curvature, henceistheround sphere.
Here comes an essential difference from the Kähler setting. The fact that compact Kähler–RiccisolitonswithpositivebisectionalcurvaturemustbeKähler–Einsteinmight simplybe acoincidence, sinceifweconsider thegeneralSasakisetting, thenacompact Sasaki–Ricci soliton with positivetransverse bisectional curvature doesnot haveto be Sasaki–Einstein.IndeedL.F. Wu([58],seealso[17])provedtheexistenceofanon-trivial RiccisolitonwithpositivecurvatureonS2withcertainorbifoldsingularity;thisimplies the existenceofanon-EinsteinSasaki–Riccisoliton onS3 withpositivetransversecur- vature. Recently Futaki–Ono–Wang [23] proved the existence of Sasaki–Ricci solitons oncompacttoricSasakimanifolds;theirresultsproducedafamilyoftoricSasaki–Ricci solitons ontheweightedSasakisphereS2n+1;thepositivityconditioncanbeassuredif aSasaki–Riccisolitoniscloseto theround metriconthesphere. ActuallyTheorem 1.2 asserts thatalltoricSasaki–Riccisolitons inthis familyhavepositivetransverse bisec- tionalcurvature.Inshort,themodelstructureintheSasakisettingisnotauniqueone, butawholefamily.
The problem isnow reducedto classifyingSasaki–Ricci solitonswith positivetrans- versebisectionalcurvature.OurstrategyistodeformsuchsolitonstoaSasaki–Einstein metric with positive transverse bisectional curvature. Note that there is no such cor- responding deformation withintheframework ofKähler geometry. Suchaflexibility in Sasakisettingseemstobeoneoftheadvantagesofthenewapproach.Thisnotonlyim- plies thataKähler–Ricci solitonwithpositivebisectionalcurvature isKähler–Einstein,
hencegives a newanalytic proof of Siu–Yau theorem,but also allowsus to generalize theresultsto theSasakisetting(Theorem 1.1).
To carry out the deformation of Sasaki–Ricci solitons, we first recall the theory of volume minimization due to Martelli–Sparks–Yau [40]. It is observed in [40] that the volume of a compact Sasaki manifold is equivalent to the Einstein-Hilbert functional, andisafunctionoftheReebvectorfieldonly.FixthecomplexstructureontheKähler cone(X,J) andamaximalcompacttorusTintheautomorphismgroupAut(X,J),they obtainedabeautifulvariationalpictureofthevolumefunctionalontheLiealgebraofT. Inparticular,thefunctionalisconvex anditscriticalpoint,ifexists,istheReebvector fieldfortheputativeSasaki–Einstein metric;moreover,thefirstvariationofthevolume functionalcanbeinterpretedastheFutakiinvariant(seealso[23]).Itisthenverynatural todeformReebvectorfieldstoreducethevolumefunctional,andtodeformSasaki–Ricci solitonscorrespondingly,withthehopetoreachacriticalReebvectorfieldwhereweend upwithadesired Sasaki–Einsteinmetric. Thisis theheuristicstrategy wetake.Along theway we alsodevelop therudimentsfor thetheory of thisnew deformation,and we hopeitwillalsobe usefulinmoregeneralsetting(cf. Section6).
Now we outline the organization of the article. InSection 2we set up various defi- nitions. In Section2.1 we giveagentle introduction to Sasaki geometry. Inparticular, werecallthenotionoftransversebisectionalcurvature. InSection2.2weintroducethe notion of a Reeb cone and a simple deformation of Sasaki structures, for ourpurpose in this paper. In Section 2.3 we introduce weighted Sasaki spheres and simple Sasaki structures on them,which form the canonicalmodels for ourstudy. InSection2.4 we recallthenotionofaSasaki–Riccisoliton.Section2.5studiesthenormalization weuse whenwedeformtheReebvectorfields.
ThemaingeometricstudyisinSection3.InSections3.1and3.2westudythevolume functional and Perelman’s μ functional. These provide a priori geometric bounds for our deformation. In Section 3.3 we carry out the above deformation picture to prove Theorems 1.1,1.2,and Corollaries 1.1,1.2,modulotechnical resultsprovedinSections 4and5.
InSection4we study thelocal property of thedeformation, using implicitfunction theoremtoprovetwotechnicalresults.Section4.1isconcernedwiththelocaldeforma- tionofSasaki–Riccisolitons,andinSection4.2weprovetherigidityofSasakimanifolds withpositivetransversebisectional curvature.Thisrigidityiscrucial heresinceincom- plex geometry one often meets the problem of jumping phenomenon. In Section 5 we studycompactnessofasequenceofSasaki–Riccisolitons withpositivetransversebisec- tionalcurvature. InSection6we discussrelatedproblems.
2. PreliminariesinSasakigeometry
Sasakigeometryhasmanyequivalentdescriptions.Wewilllargelyusetheformulation byKählercones;see,forexample,[40]foranicereference.Itcanalsobedefinedinterms ofmetriccontactgeometryortransverse Kählergeometry;see [5], forreferences.
2.1. Sasakimanifolds
Let M be acompact differentiable manifold of dimension2n+ 1(n ≥1).A Sasaki structure onM is definedto be aKähler conestructureonX =M×R+,i.e.aKähler metric(gX,J) onX oftheform
gX =dr2+r2g,
wherer >0 isacoordinateonR+,andgisaRiemannianmetriconM.Wecall(X,gX,J) theKählerconeofM.Thevertexisnotviewedaspartoftheconethroughoutthispaper.
We alsoidentifyM with thelink{r= 1}inX ifthere isnoambiguity. Becauseof the cone structure,theKählerform onX canbeexpressedas
ωX= 1 2
√−1∂∂r2=1 4ddcr2.
We denote byr∂r thehomothetic vector fieldon thecone, whichis easilyseento be a real holomorphicvectorfield.A tensor αonX issaidtobeof homotheticdegreekif
Lr∂rα=kα.
In particular, ω and g havehomothetic degree two,while J and r∂r have homothetic degreezero.WedefinetheReeb vectorfield
ξ=J(r∂r).
Then ξ is aholomorphic Killing fieldon X with homothetic degreezero. Let η be the dual one-formtoξ:
η(·) =r−2gX(ξ,·) =dclogr=√
−1(∂−∂) logr.
Wealsouse(ξ,η) todenote therestrictionofthem on(M,g).Thenwehave
• η isacontactformonM,andξisaKillingvectorfieldonM whichwealsocall the Reebvectorfield;
• η(ξ)= 1,ιξdη(·)=dη(ξ,·)= 0;
• theintegralcurvesofξ aregeodesics.
The Reeb vector field ξ defines a foliation Fξ of M by geodesics. There is aclassi- fication of Sasaki structures according to the global property of the leaves. If all the leaves are compact, then ξ generatesa circle actionon M, and theSasaki structure is called quasi-regular. In general this action is only locally free, and we get a polarized orbifoldstructure ontheleaf space.Ifthe circleaction isgloballyfree, thentheSasaki structure is called regular, and the leaf space is a polarized Kähler manifold. If ξ has
anon-compactleaftheSasaki structureis called irregular.Readers arereferredtoSec- tion2.3for examples.Inthepresentpaper theregularityof aSasaki structure willnot beessential.
Thereisanorthogonaldecompositionofthetangentbundle T M =Lξ⊕ D,
whereLξ isthe trivialbundle generalizedbyξ,and D= Ker(η). Themetricgand the contactformη determinea(1,1) tensorfieldΦ onM by
g(Y, Z) = 1
2dη(Y,ΦZ), Y, Z∈Γ(D).
Φ restrictsto analmostcomplexstructureonD: Φ2=−I+η⊗ξ.
Since both g and η are invariant under ξ, there is a well-defined Kähler structure (gT,ωT,JT) on the local leaf space of the Reeb foliation. We call this a transverse Kähler structure.In thequasi-regularcase, thisis thesameas the Kählerstructure on thequotient.Clearly
ωT = 1 2dη.
The upperscript T is used to denote both the transverse geometric quantity, and the correspondingquantityonthebundleD.ForexamplewehaveonM
g=η⊗η+gT.
From the above discussion it is not hard to see that there is an intrinsic formulation of a Sasaki structure as a compatible integrable pair (η,Φ), where η is a contact one form and Φ is an almost CR structure on D = Kerη. Here “compatible” means first that dη(ΦU,ΦV) = dη(U,V) for any U,V ∈ D, and dη(U,ΦU) > 0 for any non-zero U ∈ D. Furtherwe requireLξΦ= 0,where ξis the uniquevector fieldwith η(ξ)= 1, anddη(ξ,·)= 0.Φ inducesasplitting
D ⊗C=D1,0⊕ D0,1,
withD1,0=D0,1.“Integrable”meansthat[D0,1,D0,1]⊂ D0,1.Thisisequivalenttothat the induced almost complex structure on the local leaf space of the foliation by ξ is integrable.Formorediscussionsonthis, see[5,Chapter 6].
The Sasaki structure on M is determined by the triple (ξ,η,g).4 From now on, we will use thenotation (M,ξ,η,g) todenote aSasaki manifold. Byaneasy computation forany tangentvectorY,
R(Z, ξ)Y =g(ξ, Y)Z−g(Z, Y)ξ. (2.1) It follows that the sectional curvature of any tangent plane in M containing ξ has to be 1;orequivalently,thesectionalcurvatureofanytangentplaneinX containingeither
∂r or ξ,iszero.HenceifaSasakimanifold(M,g) ofdimension2n+ 1 isEinstein,then theEinsteinconstantmust be2n,i.e.
Ric= 2ng;
correspondingly, theKählercone(X,gX,J) isthenaRicciflatcone, i.e.
RicX=Ric−2ng= 0.
One can introduce the transverse connection ∇T and transverse curvature operator RT(Y,Z)W forY,Z, W ∈Γ(D).A straightforwardcomputationshowsthat
Ric(Y, Z) =RicT(Y, Z)−2gT(Y, Z), Y, Z ∈Γ(D).
HencetheSasaki–Einstein equationcanalsobewrittenasatransverseKähler–Einstein equation:
RicT −2(n+ 1)gT = 0. (2.2)
Weareinterestedintransverseholomorphicbisectionalcurvature.Ithasbeenstudied recently[59,31].Werecallsomedefinitions.
Definition2.1.GiventwoΦ-invarianttangentplanesσ1,σ2inDx⊂TxM,thetransverse holomorphic bisectionalcurvature HT(σ1,σ2) isdefinedas
HT(σ1, σ2) = RT(Y, J Y)J Z, Z,
where Y ∈ σ1,Z ∈ σ2 are both of unit length. We define the transverse holomorphic sectionalcurvatureof aΦ-invarianttangentplaneas
HT(σ) =HT(σ, σ).
It iseasy tocheck these arewell-defined. Forbrevity, wewill simplysay“transverse bisectional curvature”insteadof“transverseholomorphicbisectionalcurvature”.
4 Indeed,ξandηaredeterminedbyeachother,butwekeepthenotationhereinordertoemphasisboth theReebvectorfieldandthecontactform.
Definition 2.2. For x ∈ M, we say the transverse bisectional curvature is positive (or nonnegative)at xifHT(σ1,σ2) ispositiveforany two Φ invariantplanesσ1,σ2 inDx. WesayM haspositive(nonnegative)transversebisectional curvature ifHT ispositive (nonnegative)atany pointx∈M.
Itisoftenconvenienttointroducetransverseholomorphiccoordinates.Let(z1,· · ·,zn) be aholomorphicchartonalocal leafspace aroundx.Then thetransverse bisectional curvatureispositiveatxifandonlyifforanytwonon-zero tangentvectorsu=
ui ∂∂zi
andv= vj ∂∂zj,
RT(u,v)v,¯ u¯=RTi¯jk¯luiu¯jvkv¯l>0.
Thetransversebisectionalcurvaturedeterminesthetransversesectionalcurvature,so by(2.1)itdeterminesthesectionalcurvatureofM.Wehavethefollowingclassification, whichisthestartingpointofourstudy.
Lemma 2.1. A compact simply connectedSasaki–Einstein manifold with positive trans- verse bisectionalcurvature is isomorphic tothestandard Sasaki structureon S2n+1, or equivalently,theKähler coneisisometric tothestandardflat cone Cn+1\ {0}.
Proof. Thisisessentiallyaknownresult.In[31],usingthemaximumprincipleasin[15], itisprovedthatsuchmanifoldsmust haveconstanttransverse holomorphicbisectional curvature 1. Equivalently, this Sasaki structure has constant Φ-holomorphic sectional curvature 1. S. Tanno [52] has given a full classification of simply-connected Sasaki manifoldswithconstantΦ-holomorphicsectionalcurvature.Inparticular,heprovedthat asimply connected Sasaki manifolds with constantΦ-holomorphic sectional curvature 1 isisometric(as aSasakistructure) tothestandardSasaki structureon S2n+1(1),see Proposition 4.1 in [52]. While the Kähler cone corresponding to the standard Sasaki structureonS2n+1(1) is justthestandardflatcone Cn+1\ {0}. 2
2.2. Deformation ofSasaki structures
Let(M,ξ,η,g) beagivenSasakistructure.Notethatforanypositiveconstantλ= 1, the naive scaling (M,λg) is not aSasaki metric,for example, by (2.1). But there is a well-knownreplacementintheSasakisetting,called D-homothetictransformation,that isintroduced byS.Tanno [51](we shalluse homothetictransformation forsimplicity).
Itisinducedbythetransformationsξ→λ−1ξandη→λη;thecorrespondingmetricis thengivenby
gλ=λ2η⊗η+λgT.
HencethetransverseKählermetricisrescaledindeed,butthescalingfactorisdifferent from thatalongtheReebvectorfielddirection.OntheconeX, thiscanberealized by thetransformation r→r˜=rλ,and theKählerformisgivenby
˜ ω=
√−1 2 ∂∂˜r2=
√−1 2 ∂∂r2λ.
Inthepresentpaperwewillfixaparticularscalingnormalization,anditwillbespecified later (see(2.3)).
The deformations of Sasaki structures on M that we are interested in will all be inducedbyadeformationoftheKählerconemetricsonX,withafixedcomplexstruc- tureJ,i.e.adeformationoftheKählerpotentialsr2.WefirstconsidertransverseKähler deformation, as discussedin[40,23].This isaspecial caseofaType IIdeformation in- troducedin[6].Given aSasakistructure(M,ξ,η,g) anditsKählercone(X,gX,J).We consider allKähler cone metrics on(X,J) with Reebvector fieldξ. This is equivalent to fixing the homothetic vector field r∂r = −J ξ. Let r˜2/2 be the Kähler potential of another suchKähler conemetric ˜g.Thenwe have
˜
r∂˜r=r∂r.
Writing r˜2 = r2exp(2φ), the condition becomes ∂rφ = 0. Note that the Kähler cone conditionimpliesLξr˜= 0.ItfollowsthatLξφ= 0.Henceφcanalsobeconsideredasa basic function onM.Onthecone(X,J) wehave
˜
η=J(dlog ˜r) =η+dcφ.
Wesummarizethediscussion aboveas follows,
Definition2.3.Let(M,ξ,η,g) beaSasakistructureand(X,J) betheunderlyingcomplex manifold of itsKähler cone. A transverse Kähler transformation is induced by abasic function φ on M such that (X,J,ξ) remains unchanged and the Sasaki structure is inducedbythecontactformη˜=η+dcφand˜r=rexp(φ).
ItisalsoverynaturaltopresentthisdeformationintermsofbasicformsontheSasaki manifold M;see[23,5],forexample,fornicereferences.Firstwerecall,
Definition 2.4.Ap-formθ onM iscalledbasicif ιξθ= 0, Lξθ= 0.
Let ΛpB bethesheafof germsof basicp-forms andΩpB = Γ(S,ΛpB) thespaceofsmooth sectionsof ΛpB.
Theexteriordifferential preserves basicforms andwe set dB =d|ΩpB.Thus thesub- algebraΩB(Fξ) formsasubcomplexofthede Rhamcomplex,and itscohomologyring HB∗(Fξ) iscalled the basic cohomology ring. Inparticular, there is atransverse Hodge theory[20,35,57].ThetransverseHodgestaroperator∗Bisdefinedintermsoftheusual Hodgestarby
∗Bα=∗(η∧α).
Theadjointd∗B : ΩpB→Ωp−1B ofdB is
d∗B =− ∗BdB∗B.
ThebasicLaplacian operatorisdefinedtobeΔB=dBd∗B+d∗BdB.When(M,ξ,η,g) is aSasaki structure,there isanaturalsplittingofΛpB⊗Csuchthat
ΛpB⊗C=⊕Λi,jB,
whereΛi,jB isthebundleoftype(i,j) basicforms.Wethushavethewell-definedoperators
∂B : Ωi,jB →Ωi+1,jB ,
∂¯B : Ωi,jB →Ωi,j+1B . ThenwehavedB=∂B+ ¯∂B.SetdcB =√
−1∂¯B−∂B
;then dBdcB = 2√
−1∂B∂¯B, d2B= (dcB)2= 0.
ThetransverseKählerformdefines abasiccohomologyclass[ωT]B.
Nowwereturnto theabovedeformation.η˜couldbe viewedas thecontactone-form of a new Sasaki structure on M by pulling back through the embedding of M into X =M×R+ as {r˜= 1}={r=e−φ(x)}. It is straightforwardto check thatξ, η and dcφareinvariantunderthediffeomorphism
Fφ:X →X; (x, r)→(x, re−φ(x)).
Soascontactone-formsonM,wehave
˜
η=η+dcBφ.
Therefore,thetransverseKählerforms arerelatedby
˜
ωT =ωT +√
−1∂B∂Bφ.
Intheregular case,thiscorresponds todeformtheKählermetriconthequotientKäh- ler manifold within a fixed Kähler class. The transverse Ricciform ρT defines a basic cohomology class 2π1 [ρT]B,whichwecallthebasic firstChern classcB1.
Suppose now (M,ξ,η,g) has positive transverse bisectional curvature, then it fol- lows directly cB1 > 0. Moreover, a rather standard Bochner technique implies bB2 = dimHB2(M,R)= 1 (see [31]Section 9). It thenfollows thatthere isapositiveconstant λsuchthat
cB1 = 1
2π[ρT]B =λ[ωT]B.
Byahomothetictransformation,wecanthenassumethat,asabasiccohomologyclass, [ρT]B= (2n+ 2)[ωT]B. (2.3) Hencethereis abasicfunctionhonM suchthat
ρT +√
−1∂B∂Bh= (2n+ 2)ωT, (2.4) where hiscalledthe(transverse) Riccipotential ofωT,with thenormalization
M
e−hdvg = 1.
Note thath can be considered as a basicfunction on M or a function on X which is invariantunderbothξ andr∂r.OntheconeX, theRicciform ρX satisfies
ρX+√
−1∂∂h=ρT−(2n+ 2)ωT +√
−1∂B∂Bh= 0.
IftheSasaki metricisEinstein,oneeasilyseesthat(2.3)holdsandh= 0.
Next we consider more general deformations by allowing the Reeb vector field to vary in a fixed abelian Lie algebra. First we recall Type-I deformation defined in [6].
Let (M,ξ0,η0,g0) be a compact Sasaki manifold, denote its automorphism group by Aut(M,ξ0,η0,g0). We remark that here Aut(M,ξ0,η0,g0) denotes the group of all C1 diffeomorphisms ofM thatpreservesξ0,η0,g0.ElementsinAut(ξ0,η0,g0) areautomat- ically Ck+1 if g0 is inCk (k ≥1) [11]; inparticular, they are automaticallysmooth if g0 is smooth. Then Aut(M,ξ0,η0,g0) acts on (X,J) naturallyand it is asubgroup of Aut(X,J).Wefix amaximaltorus T⊂Aut(M,ξ0,η0,g0).
Definition2.5(Type-Ideformation).Let(M,ξ0,η0,g0) beaT-invariantSasakistructure and let T ⊂ Aut(M,ξ0,η0,g0) be a compact maximal torus. For any ξ ∈ t such that η0(ξ)>0.WedefineanewSasakistructureonM explicitlyas
η = η0
η0(ξ),Φ = Φ0−Φ0ξ⊗η, g=η⊗η+1
2dη(I⊗Φ). (2.5)
Itisclearfrom(2.5)thatthefamilyofSasakistructuresdepends smoothly(Ck)onthe ReebvectorfieldiftheSasakistructure(ξ0,η0,g0) issmooth(Ck).Itwasproveddirectly in[6]thatthedeformationgivenin(2.5)preservestheCRstructure(D= Ker(η0),Φ0|D).
ItturnsoutthatthereisanequivalentdescriptionofType-Ideformationonthecone (X,J).Inparticular,theunderlyingcomplexconescorrespondingtoType-Ideformation arethesame.Asin[40],wefixacompacttorusT⊂Aut(X,J) andvarytheReebvector fieldswithintheLiealgebratofT.WefixaTinvariantSasakimetric(M,ξ0,η0,g0) with ξ0∈t. Wehavethefollowing,
Lemma 2.2. For any ξ∈ t such that η0(ξ) >0, then there exists a T invariant Kähler conemetric gX on(X,J)with Reebvector fieldξ.
Proof. Let r02 be the Kähler potential on(X,J) corresponding to theSasaki structure (M,ξ0,η0,g0). To determine the required Kähler cone metric, we need to describe its radialfunction r. Wenotice −J ξ,r−10 ∂r0=η0(ξ), whichis positive andr0 invariant, henceisuniformlypositiveandbounded.Soforanyx∈X,theintegralcurveφt(x) ofthe vectorfield−J ξonX isalwaystransversetothelinkr0=constant atananglestrictly between 0 and π. In particular limt→−∞r0(φt(p)) = 0 and limt→∞r0(φt(p)) = +∞. Now we identify M with {r0 = 1} and thus M×R+ with X using the cone metric corresponding to (ξ0,η0,g0) (as in the beginning of this section). Then we obtain a diffeomorphismF : M×R+ → M×R+ which sends (p,r0) to the point φlogr0(p) = (q(p,r0),r(p,r0)).It isclearfrom thedefinition andtheaboveidentification of X with M×R+ thatwemayview(q,r) asnewcoordinatesonX,andwehaver∂r=−J ξ.We notethatrisTinvariantandlimr0→0(+∞)r= 0(+∞).Define
ω=√
−1∂∂
r2 2
andgX=ω(·, J·).
Setη=J(r−1dr).Wehave
ω=1
2d(r2η);gX=dr2+r2(η⊗η+1
2dη(I⊗J)) (2.6) Notethatthe link{r= 1}coincides with{r0= 1}.On thislink,V ∈Ker(η) if and onlyifJ V istangentto{r= 1},whichisthesameasJ V istangentto{r0= 1}.Sothis isequivalenttoV ∈Ker(η0).Itthenfollowsthaton{r0= 1},thereexistsafunctionf so thatη=f η0.Indeed,f = 1/η0(ξ)>0.WeclaimthatgX ispositivedefiniteanddefines aKählerconemetriconX.Firstitisreadilyseenthat ∂r∂ andr−1ξareorthogonaland ofunitnorm.Moreover,foranyY ∈Ker(η),gX(r∂r,Y)=gX(ξ,Y)= 0.Soitsufficesto checkgX is positivedefinite onKer(η).Note thatby(2.6),gX (andω)hashomothetic degreetwo with respect to r∂r. Thus we only needto provegX is positive definite on
Ker(η) whenrestrictedon {r= 1}.Forany two vectorsY, Z ∈Ker(η) on{r= 1},we have
dη(Y, J Z) = 1
η0(ξ)dη0(Y, J Z).
Note thatwe alsohaveLξr=dr(ξ)=dr(J(r∂r))= 0. Itfollows that(M×R+,J,gX) defines aKählercone metricwith radialfunctionrandReebsectorfieldξ.
TheconstructionaboveisindeedequivalenttoType-Ideformationgivenin(2.5).We restrict η as a 1-form on the link {r0 = 1} and we have η = η0/η0(ξ). We can then directlycomputethatdη(ξ,·)= 0 usingthefactthatη0isξ-invariant.Henceηdefinesa contact1-formon{r0= 1}withtheReebvectorfieldξ.LetΦY =J Y,Y ∈Ker(η) and Φξ= 0.Then(η,ξ,Φ) isacompatibletriple on{r0= 1}and itdefines aSasaki metric by
g=η⊗η+1
2dη(I⊗Φ).
Now weconstructaconemetricgX=dr2+r2gon(X,J) usingthefunctionr,andgX
is theKähler (cone)metricwhichcorrespondsto theSasakistructure(η,ξ,Φ). 2 Itturnsoutη0(ξ)>0 isalsonecessaryforξ∈tbeingaReebvectorfieldofacontact 1-formthatcomesoutofdeformation.
Lemma 2.3.Letη(t)(t∈[0,1])beacontinuouspath ofcontact1-forms withReeb vector fieldξ(t)∈tsuchthatη(0)=η0isthefixedT-invariantcontact1-form,thenη0(ξ(t))>0 forallt∈[0,1].
Proof. Theproperty thatη0(ξ(t))>0 isclearly anopenproperty int.So itsufficesto prove thisis also aclosed property. Thuswe canassume η0(ξ(t))>0 forall t∈[0,1), and weneedtoproveη0(ξ(1))>0.Forsimplicity wedenoteξ=ξ(1).Supposethis not true,thenbycontinuity,η0(ξ)≥0 andthereexistsapointp∈Msuchthatη0(ξ)(p)= 0.
Note thatη0∧(dη0)n and η∧(dη)n are two volumeforms onM. Then there exists a nowherevanishing functionf suchthat
f η0∧(dη0)n=η∧(dη)n. Clearly f >0.Itfollows that
(dη)n =ιξη∧(dη)n =ιξ(f η0∧(dη0)n). (2.7) Wecompute
ιξ(f η0∧(dη0)n) =f η0(ξ)(dη0)n−f η0∧ιξ(dη0)n
=f η0(ξ)(dη0)n−nf η0∧ιξdη0∧(dη0)n−1.
Notethatη0 isT-invariant;inparticular
Lξη0=ιξdη0+d(η0(ξ)) = 0.
So
ιξ(f η0∧(dη0)n) =f η0(ξ)(dη0)n+nf η0∧d(η0(ξ))∧(dη0)n−1. (2.8) Itthenfollows from(2.7)and(2.8)that
(dη)n=f η0(ξ)(dη0)n+nf η0∧d(η0(ξ))∧(dη0)n−1. (2.9) Note thatatp, η0(ξ)= 0 byassumption;also d(η0(ξ))(p)= 0 since pisaminimum of η0(ξ).By(2.9), itimpliesthat(dη)n(p)= 0. Contradiction. 2
Definition2.6.Wedefine theassociated Reebcone Rξ0 ofξ0 tobe Rξ0 ={ξ∈t:η0(ξ)>0}.
ThenotionofaReebcone originates from[40, Section 2.5],where itisdefinedtobe thecone int dualto themoment coneC∗ int∗.This isthe path-connectedcomponent of ξ0 in theset of allpossible elements in tthat are Reeb vector fieldof some Kähler conemetricon(X,J).ByLemma 2.2andLemma 2.3thenotionofaReebcone stated inDefinition 2.7 agreeswith thatgivenin[40].Thenotion ofReebcone alsocoincides withtheSasaki cone forafixed CRstructureintroduced in[7]. NotethataReebcone isalwaysconvex, andit isprovedin[40]thattheelementinRξ0 thatistheReebfield ofaSasaki–Einstein metricisunique.
Definition2.7(Simpledeformation).Wedefineasimpledeformationof(M,ξ0,η0,g0) on M withrespecttoTtobeaSasakistructurethatisinducedfromaKählerconemetric on(X,J) with Reebvector fields inRξ0. Here we emphasizethatwhen we talk about simpledeformations, we fix theunderlyingcomplex structure onthe Kähler cone, and wealsoneedtospecifyatorusT.
BytheabovediscussionweknowasimpledeformationisthecompositionofaType I deformationfollowedbyatransverseKählerdeformation.Forourpurposeweshallalso talkaboutaCk,α simpledeformationfork∈Nandα∈(0,1).
Definition2.8(Ck,a simpledeformation).ACk,αsimpledeformationisasmoothtype-I deformation(withrespecttoafixedtorusT)followedbyatransverseKählerdeformation given byaCk+2,α deformation ofthe transverseKähler potential.Notice that,by this definition,theReebvector fields involvedallbelongto theLie algebrat,so are always smooth.
Remark 2.4. It would be interesting to understand whether it is possible to have two different Reebcones forfixed(X,J) andt.
2.3. WeightedSasaki spheres
Consider thestandard Sasaki structure on thesphere S2n+1. TheKähler cone X is Cn+1\ {0}withtheflatmetricω=√−12 n
i=0dzi∧d¯zi.ThecontactformonthelinkM is givenbyη = r12
n
i=0(yidxi−xidyi),and theReebvectorfieldisξ=n
i=0(yi ∂∂xi − xi ∂∂yi).Herezi=xi+√
−1yi.TheautomorphismgroupofthisSasakimetricisU(n+1).
We take a maximal torus Tn+1 in U(n+ 1) consisting of diagonal elements. The Lie algebra t isgenerated bythe elements ξi =yi ∂∂xi −xi ∂∂yi. Forany ξ =n
i=0aiξi, we have η(ξ)=n
i=0ai|zi|2. Soforξ to be positive,itis equivalentthatai >0 for alli.
Thus theReebconeinthiscaseisRn+1+ .
We call aSasaki structure onS2n+1 simple ifit is isomorphic to aSasaki structure on S2n+1 that comes out of a simple deformation from the standard Sasaki structure (ξ,η,g).TheKählerconeofasimpleSasakistructureisbi-holomorphictoCn+1\{0}and the correspondingSasaki manifoldis called aweighted Sasaki sphere.AllsimpleSasaki structuresformaconnectedfamilyofSasakistructuresonS2n+1.Fora= (a0,· · ·,an)∈ Rn+1+ ,wedenotebyξa=
iaiξi.ItisnothardtoseethatasimpleSasakistructureon S2n+1 withReebvectorfieldξaisquasi-regularifandonlyifa∈Qn+,inwhichcasewe get acircle bundleover aweightedprojective space.Itis regularprecisely whenallthe ai’s are equal,inwhichcasethe Sasaki structureis isomorphicto the standardoneon S2n+1 uptoahomothetic transformation.
2.4. Sasaki–Riccisolitons
InthissubsectionwerecallsomegeneraltheoryonthesymmetriesinSasakigeometry, largely following [40] and [23]. We also state some facts about Sasaki–Ricci solitons, whoseproofsfollowanalogouslytheKählercaseasin[55,56].Let(M,ξ,η,g) beaSasaki manifold, and (X,gX,J) the corresponding Kählercone. Denote G= Aut(ξ,η,g),and denote bygtheLiealgebraofG.
Definition2.9.WesayavectorfieldY onaSasakimanifold(M,ξ,η,g) isaHamiltonian holomorphicvectorfield ifitshomogeneousextensiontoX (whichwealsodenotebyY) is aHamiltonianholomorphicvectorfieldwithrespectto theKählermetric(J,gX).
This definition is essentially the sameas the one given in[23], where it is phrased in terms of transverse Kähler geometry. The difference is that in[23] the vector fields are allowed to be complex valued, while inthis definitionwe onlyconsider real valued
vectorfields.HenceaHamiltonianholomorphicvectorfieldisclearlyaKillingfield.For aHamiltonian holomorphicvectorfieldY,wehaveLYr= 0 andLr∂rY = 0,then
ιYω=1
2ιYd(r2η) =1
2LY(r2η)−1
2d(r2η(Y)) =−1
2d(r2η(Y)).
SoY isgeneratedbytheHamiltonianfunctionHY =−12r2η(Y),i.e.
Y =1
2J∇X(r2η(Y)) =η(Y)ξ+r2
2J∇Xη(Y).
WecallthefunctionHY aHamiltonianholomorphicpotentialonM.Itiseasytoseethat theLiealgebragcanbeidentifiedwiththeLiealgebraofHamiltonianholomorphicvector fields Y on M, or equivalently, the Lie algebra of Hamiltonian holomorphic potentials onM underthePoissonbracket.
Now we fix the homothetic re-scaling by equation (2.3). Note this normalization is whatoneshoulduseifoneis searchingforSasaki–Einstein metrics. Lethbe theRicci potentialofωT, asdefinedin(2.4).ThereisanalternativecharacterizationofaHamil- tonianholomorphicvectorfieldintermsofaself-adjointoperatoronMwithrespectto themeasuree−hdv.Theoperatorisgiven by
L(ψ) = Δψ− ∇h· ∇ψ+ 4(n+ 1)ψ,
forabasicfunctionψ,whereΔ istheroughLaplacianofg.Notethatforabasicfunction ψ,Δψ=−ΔBψ.ThecorrespondingoperatorontheconeX isgiven by
LX(ψ) =r2(ΔXψ− ∇Xh· ∇Xψ) + 4(n+ 1)ψ.
Definition2.10.WecallabasicfunctionψonM normalized if
M
ψe−hdvg= 0. (2.10)
Foranarbitrarybasicfunctionψ,wedenote by ψ=ψ−
M
ψe−hdvg (2.11)
thenormalizationofψ.A straightforwardBochnertechniquegives
Lemma 2.5. (See[23].) A real-valuedbasic functionψ satisfies L(ψ)= 0 if andonly if ψ isaHamiltonianholomorphic potential.
Later we will also consider complex Hamiltonian holomorphic potentials, which are complex-valued basic functions ψ which satisfy that L(ψ) = 0. Similar as before, the
space ofallcomplexHamiltonianholomorphic potentialscanbeidentified withtheLie algebra of the group P of holomorphic transformations of (X,J) that commute with thedilationgeneratedbyr∂r.Inliterature,P isoftencalledthetransverseholomorphic automorphismgroup.ClearlyPdoesnotchangeunderthetransverseKählerdeformation of theSasaki structure.
Forany Y ∈g,wedefinetheFutakiinvariant ofJ Y as Fut(J Y) =
X
LJ Y(−h)e−r
2
2dVX =1 2
X
∇Xh· ∇Xη(Y)r2e−r
2 2dVX.
Theappearanceoftheexponentialtermguaranteesthevalidityofintegrationbyparts, and onecanshowthatthis doesnotchangeunderatransversal Kählerdeformation of themetric.
Now weintroducethenotionofaSasaki–Riccisoliton,following [23].
Definition2.11.AcompactSasakimanifold(M,ξ,η,g) iscalled aSasaki–Riccisolitonif there isaHamiltonian holomorphicvectorfieldY suchthat
RicT −(2n+ 2)gT =LJ YgT.
ThisisequivalenttothattheRiccipotentialhisaHamiltonianholomorphicpotential, inotherwords,L(h)= 0.SupposeTisamaximaltorusofGand(M,ξ,η,g) isaSasaki–
Riccisoliton,thenitisSasaki–EinsteinifFut(J Y)= 0 forallY ∈t.Indeed,theHamil- tonianvectorfieldYh generatedbytheRiccipotentialhcommuteswithallelementsin t, and bymaximality of T, it must be in t. So Fut(J Yh) = 12
X|∇Xh|2e−r22dVX = 0, and thushisconstant.
2.5. Thenormalization
By definition a Sasaki–Ricci soliton satisfies the normalization condition (2.3), and this fixesthehomothetictransformationoftheReebvectorfield.Thuswhenwedeform Sasaki–Riccisolitons,onewouldliketoaskunderwhatconditiononξis(2.3)preserved.
Definition2.12.WesayacompactSasakimanifold(M,ξ,η,g) isnormalizedifitsatisfies (2.3).i.e.
2πcB1 = (2n+ 2)[ωT].
By[23] aSasaki manifoldis homothetic toanormalized oneif andonly ifthebasic first ChernclasscB1 ispositivedefiniteandthecontactsubbundleDhasvanishingfirst Chernclass.WehavealreadyseeninSection2.2thatif(M,ξ,η,g) haspositivetransverse bisectional curvature,thenitcanbenormalizedbyahomothetic transformation.
