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3 1 0 EXTREMAL KA¨HLER METRICS ON BLOW-UPS OF PARABOLIC 2 RULED SURFACES n a J CARLTIPLER 1 2 ] G ABSTRACT. NewexamplesofextremalKa¨hlermetricsaregivenonblow- D upsof parabolicruledsurfaces. Themethod usedisbased onthegluing . constructionofArezzo,PacardandSinger[4].Thisenablestoendowruled h surfacesoftheformP(O⊕L)withspecialparabolicstructuressuchthat t a theassociatediteratedblow-upadmitsanextremalmetricofnon-constant m scalarcurvature. [ 3 v 1. INTRODUCTION 5 In this paper is adressed the problem of existence of extremal Ka¨hler met- 1 3 ricsonruledsurfaces. AnextremalKa¨hlermetriconacompact Ka¨hlermani- 4 foldM isametricthatminimizestheCalabifunctionalinagivenKa¨hlerclass . 4 Ω: 0 1 1 {ω ∈Ω1,1(M,R),dω = 0, ω > 0 /[ω] = Ω} → R : v ω 7→ s(ω)2ωn. i M X R Here, s(ω) stands for the scalar curvature of ω and n is the complex di- r a mension of M. Constant scalar curvature metrics are examples of extremal metrics. If the manifold is polarized by an ample line bundle L the existence ofsuchametricintheclassc (L)isrelatedtoanotionofstability ofthepair 1 (M,L). Moreprecisely,theworksofYau[32],Tian[30],Donaldson[11]and lastlySze´kelyhidi[28],ledtotheconjecturethatapolarizedmanifold(M,L) admitsanextremalKa¨hlermetricintheKa¨hlerclassc (M)ifandonlyifitis 1 relatively K-polystable. Sofarit has been proved that the existence of a con- stantscalarcurvature Ka¨hlermetricimpliesK-stability [20]andtheexistence ofanextremalmetricimpliesrelativeK-polystability [29]. Date:June2011. 2000MathematicsSubjectClassification. Primary53C55;Secondary32Q26. 1 2 CARLTIPLER Wewillfocusonthespecialcaseofcomplexruledsurfaces. Firstconsider ageometrically ruledsurface M. Thisisthetotalspaceofafibration P(E) → Σ whereE isaholomorphic bundleofrank 2onacompactRiemannsurface Σ. In that case, the existence of extremal metrics is related to the stability of the bundle E. Alot of work has been done in this direction, werefer to [5] fora surveyonthistopic. Moreover,inthispaper,Apostolov,Calderbank,Gauduchon andTønnesen- Friedman prove that if the genus of Σ is greater than two, then M admits a metricofconstantscalarcurvatureinsomeclassifandonly ifE ispolystable. Another result due to Tønnesen-Friedman [31] is that if the genus of Σ is greater than two, then there exists an extremal Ka¨hler metric of non-constant scalar curvature on M ifand only ifM = P(O⊕L)with L aline bundle of positivedegree(seealso[27]). Notethatinthatcasethebundleisunstable. The above results admit partial counterparts in the case of parabolic ruled surfaces (see definition 1.0.1). In the papers [22] and [23], Rollin and Singer showed that the parabolic polystability of aparabolic ruled surface S implies theexistenceofaconstantscalarcurvaturemetriconaniteratedblow-upofS encoded bytheparabolic structure. It is natural to ask for such a result in the extremal case. If there exists an extremal metric of non-constant scalar curvature on an iterated blow-up of a parabolic ruled surface, theexistence oftheextremal vectorfieldimplies that M is of the form P(O ⊕ L). Moreover, the marked points of the parabolic structure mustlie onthe zero orinfinity section ofthe ruling. Inspired bythe results mentioned above, we can ask if for every unstable parabolic structure on a minimal ruled surface of the form M = P(O⊕L), with marked points on the infinity section of the ruling, one can associate an iterated blow-up of M supporting anextremalKa¨hlermetricofnon-constant scalarcurvature. Arezzo, Pacard and Singer, and then Sze´kelyhidi, proved that under some stability conditions, onecanblow-upanextremalKa¨hlermanifold andobtain an extremal Ka¨hler metric on the blown-up manifold for sufficiently small metric on the exceptional divisor. Thisblow-up process enables toprove that many of the unstable parabolic structures give rise to extremal Ka¨hler metrics of non-constant scalar curvature on the associated iterated blow-ups. A modification of their argument will enable to get more examples of extremal metricsonblow-upsencoded byunstable parabolic structures. Inordertostatetheresult, weneedsomedefinitions aboutparabolic structures. LetΣbeaRiemannsurface and Mˇ ageometrically ruled surface, total EXTREMALKA¨HLERMETRICSONBLOW-UPSOFPARABOLICRULEDSURFACES 3 spaceofafibration π : P(E) → Σ withE aholomorphic bundle. Definition1.0.1. Aparabolic structure P on π : Mˇ = P(E) → Σ is the data of s distinct points (A ) on Σ and for each of these points i 1≤i≤s the assignment of a point B ∈ π−1(A ) with a weight α ∈ (0,1) ∩ Q. i i i A geometrically ruled surface endowed with a parabolic structure is called a parabolic ruledsurface. In the paper [22], to each parabolic ruled surface is associated an iterated blow-up Φ :Bl(Mˇ,P) → Mˇ. Wewilldescribetheprocesstoconstruct Bl(Mˇ,P)inthecaseofaparabolic ruled surface whose parabolic structure consists of asingle point, the general casebeingobtainedoperatingthesamewayforeachmarkedpoint. LetMˇ → Σ be such a parabolic ruled surface with A ∈ Σ, marked point Q ∈ F := p π−1(A)andweightα= ,withpandqcoprimeintegers, 0 < p < q. Denote q p q−p theexpansions of and intocontinuous fractionsby: q q p 1 = q 1 e − 1 1 e −... 2 e k and q−p 1 = . q 1 e′ − 1 1 e′ −... 2 e′ l Suppose that the integers e and e′ are greater orequal than twoso that these i i expansions areunique. Thenfrom[22]thereexistsauniqueiteratedblow-up Φ :Bl(Mˇ,P) → Mˇ withΦ−1(F)equaltothefollowingchainofcurves: −e1 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19) −e2 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19)_ _ _(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)−ek−1(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −ek (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −1 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −e′l (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)−e′l−1(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)_ _ _(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −e′2 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19) −e′1 4 CARLTIPLER Theedgesstandfortherationalcurves,withself-intersectionnumberabove them. The dots are the intersection of the curves, of intersection number 1. Moreover, the curve of self-intersection −e is the proper transform of the 1 fiberF. Inordertogetthisblow-up,startbyblowing-upthemarked pointand obtainthefollowingcurves: −1 −1 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) . Here the curve on the left is the proper transform of the fiber and the one on therightisthefirstexceptionaldivisor. Thenblow-uptheintersectionofthese twocurvestoobtain −2 −1 −2 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) . Then choosing one of the two intersection points that the last exceptional di- visorgivesanditerating theprocess, oneobtainthefollowingchainofcurves −e1 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19) −e2 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19)_ _ _(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)−ek−1(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −ek (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −1 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −e′l (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)−e′l−1(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20)_ _ _(cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) −e′2 (cid:23)(cid:16)(cid:22)(cid:17)(cid:21)(cid:18)(cid:20)(cid:19) −e′1 Remark 1.0.2. The chain of curves on the left of the one of self-intersection number −1 is the chain of a minimal resolution of a singularity of A type p,q andtheoneontherightofasingularity of A type(seesection2). q−p,q Remark1.0.3. In[22],thecurve ofself-intersection −e isthepropertrans- 1 formoftheexceptionaldivisorofthefirstblow-upwhileherethisistheproper transform ofthefiberF. Recallthatthezerosectionofaruledsurface P(O⊕L)isthesectiongiven by the zero section of L → Σand the inclusion L ⊂ P(O⊕L). The infinity sectionisgivenbythezerosectionofO → ΣintheinclusionO ⊂ P(O⊕L). Given asurface Σ, K stands forits canonical bundle and if A ∈ Σ, [A] isthe linebundle associated tothedivisorA. Thenwecanstate: Theorem A. Let r and (q ) be positive integers such that for each j, j j=1..s q ≥ 3and j gcd(q ,r)= 1. j Foreachj,let p +r j p ≡ −r[q ], 0 < p < q , n = . j j j j j q j LetΣbeaRiemannsurfaceofgenusgandsmarkedpoints(A )onit. Define j aparabolic structure P on Mˇ = P(O⊕(Kr ⊗ [A ]r−nj)) j j EXTREMALKA¨HLERMETRICSONBLOW-UPSOFPARABOLICRULEDSURFACES 5 consisting ofthepoints(B )intheinfinitysectionoftherulingofMˇ overthe j points(A )together withtheweights(pj). If j qj 1 χ(Σ)− (1− )< 0 q Xj j then there exists an extremal Ka¨hler metric of non-constant scalar curvature onBl(Mˇ,P). Thismetricisnotsmalloneveryexceptional divisor. Remark1.0.4. Theparabolic structureisunstable. Wewillseethattheinfin- itysectiondestabilises theparabolic surface. Remark1.0.5. TheKa¨hlerclasses oftheblow-up whichadmitstheextremal metriccanbeexplicitlycomputed;thiswillbeexplainedinSection3.5. More- over, these classes are different from the one that could be obtained from the workofArezzo,PacardandSinger. Usingaslightlymoregeneralconstruction, wewillobtain: Theorem B. LetM = P(O⊕L)bearuled surface overaRiemannsurface ofgenus g,withLalinebundle ofdegree d. Ifg ≥ 2wesuppose d = 2g−2 or d ≥ 4g − 3. Then there exists explicit unstable parabolic structures on M such that each associated iterated blow-up Bl(M,P) admits an extremal Ka¨hlermetricofnon-constant scalar curvature. TheKa¨hlerclassobtained is notsmalloneveryexceptional divisor. Remark 1.0.6. In fact, a combination of the results in [28] and [29] shows that the extremal Ka¨hler metrics obtained by Tønnesen-Friedman in [31] lie exactly in the Ka¨hler classes that give relatively stable polarizations. Thus the unstable parabolic structures obtained might be in fact ”relatively stable“ parabolic structures, inasensethatremainstobeunderstood. ^ 1.1. Example. ConsiderCP1×T2athreetimesiteratedblow-upofthetotal spaceofthefibration CP1×T2 → T2. Theconsidered blow-upcontains thefollowingchainofcurves: E1 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) E2 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) E3 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) F −2 −2 −1 −3 HereF isthepropertransform ofafiberoftheruling CP1×T2 → T2. E , E and E are the exceptional divisors of the iterated blow-ups. Let S 1 2 3 0 bethepropertransform ofthezerosection. Then: 6 CARLTIPLER Theorem C. Foreach (a,b) ∈ R∗,2 such that a < k wherek isaconstant + b 2 2 defined in[31]andforeach(a ,a ,a )positive numbers, thereexists ε > 0 1 2 3 0 such that for every ε ∈ (0,ε ), there exists an extremal metric ω with non- 0 ^ constant scalarcurvature onCP1×T2. Thismetricsatisfies 2 [ω]·S = πb, 0 3 [ω]·F = ε2a , [ω]·E = ε2a , [ω]·E = ε2a 1 1 2 2 3 and π [ω]·E = (b−a) . 3 3 1.2. Strategy. TheTheorem Awillbeobtained from ageneral process. The firststepistoconsiderKa¨hlerorbifoldsendowedwithextremalmetrics. Such orbifolds can be obtained from the work of Bryant [7] and Abreu [1] on weighted projective spaces. Legendre also provide examples inthe toric case [18]. Other examples will come from the work of Tønnesen-Friedman [31], generalizedtotheorbifoldsetting. TheseorbifoldswillhaveisolatedHirzebruch- Jungsingularities. TheworkofJoyceandthenCalderbankandSingerenables ustoendowalocalmodelofresolutionofthesesingularities withascalar-flat Ka¨hler metric [10]. Then the gluing method of Arezzo, Pacard and Singer [4]isusedtogluethesemodelstotheorbifoldsandobtainmanifoldswithex- tremalKa¨hlermetrics. Notethatthereexistsanimprovementofthearguments of[4]bySze´kelyhidi in[26]. Remark1.2.1. Thegluingprocessdescribedin[4]worksinhigherdimension but there are no such metrics on local models of every resolution of isolated singularities in higher dimension. However, Joyce constructed ALE scalar- flatmetricsonCrepantresolutions[16]. Thenonecanexpecttogeneralizethe process described belowinsomecasesofhigherdimension. 1.3. Second example. Using this gluing method we obtain an other simple example. Let us consider CP2 the three-times iterated blow-up of CP2 with thefollowingchainofcurves: g H (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) E3 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) E2 (cid:16)(cid:23)(cid:17)(cid:22)(cid:18)(cid:21)(cid:19)(cid:20) E1 −2 −1 −2 −2 Here H denotes the proper transform of a line in CP2 on which the first blown-uppointlies. E ,E andE standforthepropertransformofthefirst, 1 2 3 second and last exceptional divisors. Thedots represent the intersections and the numbers below the lines are the self-intersection numbers. Then we can state: EXTREMALKA¨HLERMETRICSONBLOW-UPSOFPARABOLICRULEDSURFACES 7 TheoremD. Foreverya,a ,a ,a positivenumbersthereexistsε > 0such 1 2 3 0 that for every ε ∈ (0,ε ), there is an extremal Ka¨hler metric ω of non- 0 ε constant scalarcurvature onCP2 satisfying g[ωε]·H = ε2a3, [ω ]·E = a, [ω ]·E = ε2a ε 3 ε 2 2 and [ω ]·E = ε2a . ε 1 1 Remark 1.3.1. If one starts with the first Hirzebruch surface endowed with theCalabimetric andusetheworkofArezzo, PacardandSingertoconstruct extremalmetricson CP2,theKa¨hlerclassesobtained areoftheform g [ωε]·H = b, [ω ]·E = ε2a , [ω ]·E = ε2a ε 3 3 ε 2 2 and [ω ]·E = a. ε 1 with a and b positive real numbers and ε small enough . The Ka¨hler classes obtained with the new process can be chosen arbitrarily far from the one ob- tainedbyArezzo,PacardandSinger. 1.4. Plan of the paper. In the section 2 we set up the general gluing theorem for resolutions following [2] and [4]. In section 3 we build the orbifolds with extremal metrics that weuse in the gluing construction, and identify the surfaces obtained after resolution. This will prove Theorem A. Then in the section 4 we discuss unstable parabolic structures and give the proof of the Theorem B. In the last section we show how to obtain the examples of the introduction. Acknowledgments. I’d like to thank especially my advisor Yann Rollin for his help and encouragement. I am grateful to Paul Gauduchon and Michael Singerforallthediscussionswehad. I’dalsoliketothankVestislavApostolov who pointed to me Abreu’s and Legendre’s work. I thank Frank Pacard and Gabor Sze´kelyhidi for their remarks on the first version of this paper, as well astherefereewhosecommentsenabledtoimprovethepaper. Andlastbutnot least a special thank to Andrew Clarke for all the time he spend listening to meandallthesuggestions hemadetoimprovethispaper. 2. HIRZEBRUCH-JUNG SINGULARITIES AND EXTREMAL METRICS The aim of this section is to present the method of desingularization of extremalKa¨hlerorbifolds. 8 CARLTIPLER 2.1. Local model. We first present the local model which is used to resolve thesingularities. Definition2.1.1. Letpandqbecoprimenon-zerointegers,withp < q. Define the group Γ to be the multiplicative subgroup of U(2) generated by the p,q matrix 2iπ exp 0  (cid:18) q (cid:19)  γ := 2iπp  0 exp   (cid:18) q (cid:19)    Thegroup Γ actsonC2 : p,q 2iπ 2iπp ∀(z ,z )∈ C2,γ.(z ,z ) := exp ·z ,exp ·z . 0 1 0 1 0 1 (cid:18) (cid:18) q (cid:19) (cid:18) q (cid:19) (cid:19) Definition 2.1.2. Let p and q be coprime non zero integers, with p < q. An A singularity is a singularity isomorphic to C2/Γ . A singularity of p,q p,q Hirzebruch-Jung typeisanysingularity ofthistype. We recall some results about the resolutions of these singularities. First, from the algebraic point of view, C2/Γ is a complex orbifold with an iso- p,q latedsingularity at0. Thereexistsaminimalresolution π : Y → C2/Γ p,q p,q called the Hirzebruch-Jung resolution. The manifold Y is a complex sur- p,q face with exceptional divisor E := π−1(0) and π is a biholomorphism from Y − E to C2 − {0}/Γ. For more details about resolutions see [6]. Next, p,q C2/Γ and Y are toric manifolds. The action of the torus T2 is the one p,q p,q thatcomesfromthediagonal action onC2. Formoredetails onthisaspect of theresolutionsee[12]. Lastly,theminimalresolutioncanbeendowedwithan ALEscalar-flatKa¨hlermetricω ineachKa¨hlerclassasconstructedbyJoyce, r CalderbankandSingerin[10]. Theexceptionaldivisoroftheresolutionisthe union of CP1s embedded in Y and the volume of each of these curves can p,q be chosen arbitrarily. This metric is T2-invariant and its behaviour at infinity iscontrolled: Proposition 2.1.3. ([22],Corollary 6.4.2.) Intheholomorphic chart C2−{0}/Γ p,q themetricω isgivenbyω = ddcf,with r r 1 f(z)= |z|2+alog(|z|2)+O(|z|−1) 2 anda ≤ 0. EXTREMALKA¨HLERMETRICSONBLOW-UPSOFPARABOLICRULEDSURFACES 9 2.2. Thegluingmethod. Thegluingmethodpresentedherecomesfrom[4]. Let (M,J,ω) be a Ka¨hler orbifold with extremal metric. Suppose that the singular points of M are isolated and of Hirzebruch-Jung type. Denote by p i the singular points of M and B(p ,ε) := B(p ,ε)/Γ orbifold balls around i i i the singularities of radius ε with respect to the metric of M. Fix r > 0 0 such that the B(p ,ε) are disjoint for ε < r . Consider, for 0 < ε < r , i 0 0 the manifold M := M −∪B(p ,ε) . Let Y stand for a local model of the ε i i resolutionofthesingularityp ,endowedwiththemetricofJoyce-Calderbank- i Singer. The aim is to glue the Y to M in order to obtain a smooth Ka¨hler i ε manifold M which resolves M and has an extremal Ka¨hler metric. To do this, one needs to perturb the Ka¨hler potentials of the metrics to make them f agreeontheboundariesofthedifferentpieces,keepingtheextremalcondition on these potentials. If we consider small enough ε, the metric will look like the euclidian metric in holomorphic chart because it is Ka¨hler. On the other hand,theJoyce-Calderbank-Singer metricisALEsoonecanhopetogluethe metricstogetherwithaslightperturbation. Letsbethescalarcurvature of ω. Definetheoperator: P : C∞(M) → Λ0,1(M,T1,0) ω f 7→ 1∂Ξf 2 with Ξf = J∇f +i∇f. A result of Calabi asserts that a metric is extremal if and only if the gradi- ent field of the scalar curvature is a real holomorphic vector field. Therefore a metric ω′ is extremal if and only if Pω′(s(ω′)) = 0, with s(ω′) denoting the scalar curvature. Let P∗ be the adjoint operator of P . We will use the ω ω followingproposition: Proposition2.2.1. [19]Ξ ∈ T1,0isaKillingvectorfieldwithzerosifandonly ifthereisarealfunctionf solutionofP∗P (f)= 0suchthatω(Ξ,.) = −df. ω ω This result is initially proved for manifolds but the proof extends directly to orbifolds with isolated singularities, working equivariently in the orbifold charts. AresultofCalabi([9])statesthattheisometrygroupofanextremalmetric isamaximalcompactsubgroupofthegroupofbiholomorphismsofthemani- fold. ThusinthegluingprocesswecanprescribeacompactsubgroupT ofthe groupofbiholomorphisms ofM tobecomeasubgroup oftheisometrygroup ofM and work T-equivariantly. Wewantthisgroup T tobecontained inthe isometry group of M because the metric that will be obtained on M will be f f 10 CARLTIPLER neartotheoneonM awayfrom theexceptional divisors. Moreover, itsalge- bramustcontaintheextremalvectorfieldof ω forthesamereason. Let K be the sugroup of Isom(M,ω) consisting of exact symplectomorphisms. Let T beacompact subgroup of K such thatitsLiealgebra tcontains X := J∇s, s the extremal vector field of the metric. Let h be the Lie algebra of real- holomorphic hamiltonian vector fields which are T-invariant. These are the vector fields that remains in the T-equivariant setup. X is contained in h. h s splitsash′⊕h′′ withh′ = h∩t. Thedeformationsofthemetricmustpreserve theextremalcondition soweconsiderdeformations f 7−→ ω +i∂∂f e suchthat (2.1) −ds(ω+i∂∂f)= (ω+i∂∂f)(X +Y, .) s with Y ∈ h′. As X +Y ∈ h′, the proposition 2.2.1 of Lichnerowicz above s ensuresthatthesedeformationsareextremal. Moreover,thevectorfieldsfrom h′ areprecisely theonesthatgiveextremaldeformations. Inorderto obtain such deformations, consider the momentmapξ associ- ω atedtotheactionofK: ξ :M → k∗. ω ξ isdefinedsuchthatforevery X ∈ kthefunction hξ ,XionM isahamil- ω ω tonianforX,wichmeans ω(X, .)= −dhξ ,Xi. ω Moreover, ξ isnormalized suchthat ω hξ ,Xiωn = 0. ω Z M Theequation (2.1)cannowbereformulated: s(ω+i∂∂f)= hξ ,X +Yi+constant. ω+i∂∂f s IfweworkT-equivariantly, weareinterested intheoperator: F : h×C∞(M)T ×R → C∞(M)T (X,f,c) 7→ s(ω+i∂∂f)−hξ ,Xi−c−c ω+i∂∂f s where C∞(M)T stands forthe T-invariant functions and c isthe average of s thescalarcurvatureof ω. ThereisaresultwhichisduetoCalabiandLebrun- Simanca: