Deformations of constant scalar curvature Sasakian metrics and K-stability
aa r X i v : . [ m a t h . DG ] J a n DEFORMATIONS OF CONSTANT SCALAR CURVATURESASAKIAN METRICS AND K-STABILITY
CARL TIPLER AND CRAIG VAN COEVERING
Abstract.
Extending the work of G. Sz´ekelyhidi and T. Br¨onnle to Sasakianmanifolds we prove that a small deformation of the complex structure of thecone of a constant scalar curvature Sasakian manifold admits a constant scalarcurvature structure if it is K-polystable. This also implies that a small defor-mation of the complex structure of the cone of a constant scalar curvaturestructure is K-semistable. As applications we give examples of constant scalarcurvature Sasakian manifolds which are deformations of toric examples, andwe also show that if a 3-Sasakian manifold admits a non-trivial transversalcomplex deformation then it admits a non-trivial Sasaki-Einstein deformation. Introduction
The existence of a K¨ahler-Einstein metric on a compact K¨ahler manifold X with c ( X ) > X . G. Tian laterintroduced the notion of K-stability [41] and showed that it is a necessary conditionfor the existence of a K¨ahler-Einstein metric. Then S. Donaldson extended the no-tion of K-stability to any K¨ahler manifold whose K¨ahler class is c ( L ) for an ampleline bundle L . The Yau-Tian-Donaldson conjecture states that the existence of aconstant scalar curvature K¨ahler (cscK) metric in c ( L ) should be equivalent to theK-stability, or K-polystablity when there are non-trivial holomorphic vector fields,of the polarized variety ( X, L ). It is known from the work of S. Donaldson [12],J. Stoppa [39] and T. Mabuchi [29] that the existence of a cscK metric impliesK-stability.Since a Sasakian manifold is essentially an odd dimensional analogue of a K¨ahlermanifold, and furthermore the category of polarized K¨ahler manifolds ( X, L ) em-beds in the former as the class of regular Sasakian manifold, it is natural to makea similar conjecture for Sasakian manifolds.Besides its similarity with K¨ahler geometry interest in Sasakian manifolds hashad two main motivations. First, the work in Sasaki-Einstein manifolds in thelast fifteen years has been very prolific in producing examples of positive scalarcurvature Einstein manifolds. See for example [6, 5, 26], and the recent survey [38]and its references, since the literature is too vast to list. The second impetus hasbeen from theoretical physics with AdS/CFT correspondence which, in the mostinteresting dimensions, provides a duality between field theories and string theorieson AdS × M where supersymmetry requires the five dimensional Einstein manifold M to have a real Killing spinor (see [30, 25, 2, 32]).A polarization of a Sasakian manifold M is given by a Reeb vector field ξ andthe cone Y = ( C ( M ) , ξ ) is an affine variety polarized by ξ . T. Collins and G. Sz´ekelyhidi [42] defined the notion of a test configuration for a polarized affine vari-ety (
Y, ξ ) and were able to define K-polystability by extending the Futaki invariantto singular polarized affine varieties. They proved that existence of a constantscalar curvature Sasakian (cscS) metric implies K-semistability. This generalizesearlier work of J. Ross and R. Thomas [36] which defined K-stability for polarizedK¨ahler orbifolds and proved that the existence of a cscK orbifold metric impliesK-semistability. Collins and Sz´ekelyhidi extended this result to irregular Sasakianmanifolds.There are already well known obstructions to the existence of a Sasaki-Einsteinor cscS metric on a polarized Sasakian manifold. There is the Futaki invariant [7, 20]which is defined just as for K¨ahler manifolds, and there are the volume minimizationresults of Martelli, Sparks and Yau [31]. Also there are the Lichnerowicz andBishop obstructions of [21], which are only non-trivial for non-regular Sasakianmanifolds. But most of the research in Sasaki-Einstein manifolds has concentratedon proving existence of examples using sufficient conditions which are probably farfrom necessary. However the above results make the following conjecture natural.
Conjecture 1.
Let ( M, ξ ) be a Sasakian manifold polarized by the Reeb vector field ξ with polarized affine cone ( Y, ξ ) . Then there exists a cscS structure compatiblewith ξ and the fixed complex cone Y = C ( M ) if and only if ( Y, ξ ) is K-polystable. This article is concerned with a local study of stability. That is given a cscS man-ifold (
M, ξ, g ) with its polarized affine cone (
Y, ξ ), we consider small deformationsof the complex structure ( Y t , ξ ) t ∈U preserving ξ . We use the contact perspectiveof W. He [23], which is analogous to the study of K¨ahler structures by fixing thesymplectic form and varying the almost complex structure due to S. Donaldson [10].Thus we fix a contact manifold ( M, η, ξ ) and consider the space K of (1 , transversal almost complex structure of a structure ( g Φ , η, ξ, Φ) whichis Sasakian if Φ is integrable. The action of the exact contactomorphism group G is then Hamiltonian with moment map µ : K → C ∞ b ( M ) ,µ (Φ) = s − s the scalar curvature of g Φ and C ∞ b ( M ) are smooth functions invariantunder the Reeb flow with zero average.If one fixes a transversal complex structure J one gets the space of compatibleSasakian structures S ( ξ, ¯ J ), which is analogous to the space of K¨ahler metrics in afixed K¨ahler class. We are able to prove the following. Theorem 2.
Let ( M, η, ξ, Φ ) be a cscS manifold and ( M, η, ξ, Φ) a nearby Sasakianmanifold with transverse complex structure ¯ J . Then if ( M, η, ξ, Φ) is K-polystable,there is a constant scalar curvature Sasakian structure in the space S ( ξ, ¯ J ) . The proof of Theorem 2 uses a technique of G. Sz´ekelyhidi [40] of constructing afinite dimensional slice to the action of the “complexification” G C of the contacto-morphism group, and reducing the problem to one of finite dimensional geometricinvariant theory. T. Br¨onnle [8] also obtained similar results independently.An application of this technique is in determining when a small deformation J ofthe transversal complex structure of a cscS metric ( M, η, ξ, Φ ) admits a compatiblecscS structure in S ( ξ, ¯ J ). This is reduced to the polystability of the correspondingorbit of G G , G = Aut( η, ξ, Φ ), on H ( B • ) where B • is the appropriate deformation EFORMATIONS OF CSCS METRICS 3 complex. This problem has been considered by the second author using analyticmethods in [45, 44].A consequence of this technique is the following result extending the result of [42].
Theorem 3.
Let ( M, η, ξ, Φ ) be a cscS manifold. Then a small deformation ( M, η, ξ, Φ) fixing the Reeb vector field is K-semistable. In the final section we give some examples of toric cscS manifolds admittingpolystable deformations and thus cscS deformations, and also non-polystable de-formations. We also show that “real” deformations of a 3-Sasakian manifold arepolystable. This shows that any 3-Sasakian manifold with non-trivial deformationsof the transversal complex structure admits a non-trivial Sasaki-Einstein deforma-tion.1.1.
Acknowledgments.
This work has been presented for the first time at the18th International Symposium on Complex Geometry at Sugadaira. The first au-thor would like to thank the organizers for their invitation and hospitality. Theauthors want to thank Professors Charles Boyer, Akito Futaki, Tam´as K´alm´an,Toshiki Mabuchi and Christina Toennesen-Friedman for their interest in this work.We would also like to thank the reviewer for helpful comments which led to arevision of Proposition 3.3.1.2.
Background on Sasakian manifolds, canonical metrics andK-stability
Sasakian structures.
First we recall the definition of a Sasakian manifoldand basics of Sasakian geometry. We refer the reader to the monograph [4] for moredetails.
Definition 2.1.1.
A Riemannian manifold ( M, g ) is a Sasakian manifold if themetric cone ( C ( M ) , g ) = ( R > × M, dr + r g ) is K¨ahler with respect to somecomplex structure I , where r is the usual coordinate on R > . Thus the dimension n of M is odd and denoted n = 2 m + 1, while C ( M ) is acomplex manifold with dim C C ( M ) = m + 1.In the following we will characterize Sasakian manifolds as a special type ofmetric contact structure. We will identify M with the submanifold { } × M ⊂ C ( M ). Let r∂ r be the Euler vector field on C ( M ), we define a vector field tangentto M by ξ = Ir∂ r . Then r∂ r is real holomorphic, ξ is Killing with respect to both g and g , and furthermore the orbits of ξ are geodesics on ( M, g ). Define η = r ξ y g ,then we have(1) η = − I ∗ drr = d c log r, where d c = √− ∂ − ∂ ). If ω is the K¨ahler form of g , i.e. ω ( X, Y ) = g ( IX, Y ),then L r∂ r ω = 2 ω which implies that(2) ω = 12 d ( r∂ r y ω ) = 12 d ( r η ) = 14 dd c ( r ) . From (2) we have(3) ω = rdr ∧ η + 12 r dη. C. TIPLER AND C. VAN COEVERING
We will use the same notation to denote η and ξ restricted to M . Then (3) impliesthat η is a contact form with Reeb vector field ξ , since η ( ξ ) = 1 and L ξ η = 0. Then( M, η, ξ ) is a contact manifold.Let D ⊂ T M be the contact distribution which is defined by(4) D x = ker η x for x ∈ M . Furthermore, if we restrict the almost complex structure to D , J := I | D ,then ( D, J ) is a strictly pseudoconvex CR structure on M . There is a splitting ofthe tangent bundle T M (5)
T M = D ⊕ L ξ , where L ξ is the trivial subbundle generated by ξ . Define a tensor Φ ∈ End(
T M )by Φ | D = J and Φ( ξ ) = 0. Then we denote the Sasakian structure by ( g, η, ξ, Φ).Definition 2.1.1 is the simplest definition of a Sasakian structure, but a Sasakianstructure is also frequently defined as a metric contact structure satisfying an ad-ditional normality condition. We give some details that are needed in this articlebut for more details see [4].Assume for now that we merely have a contact manifold (
M, η, ξ ), with Reebvector field ξ . Definition 2.1.2. A (1 , -tensor field Φ :
T M → T M on a contact manifold ( M, η, ξ ) is called an almost contact-complex structure if Φ ξ = 0 , Φ = − Id + ξ ⊗ η. An almost contact-complex structure is called
K-contact if in addition, L ξ Φ = 0 . An almost contact-complex structure Φ is compatible with a metric g if g (Φ X, Φ Y ) = g ( X, Y ) − η ( X ) η ( Y ) , for X, Y ∈ Γ( T M ) . If g is a compatible metric and g (Φ X, Y ) = 12 dη ( X, Y ) , for X, Y ∈ Γ( T M ) , then ( g, η, ξ, Φ) is a called a metric contact structure . If in addition the Reeb vectorfield ξ is Killing, then it is called a K-contact metric structure . Definition 2.1.3.
An almost contact-complex structure Φ on a contact manifold ( M, η, ξ ) is compatible with η if dη (Φ X, Φ Y ) = dη ( X, Y ) , and dη ( X, Φ X ) > for X ∈ ker η, X = 0 . If Φ is compatible with η , then one defines a Riemannian metric by g Φ ( X, Y ) = 12 dη ( X, Φ Y ) + η ( X ) η ( Y ) , and ( η, ξ, Φ , g Φ ) is a contact metric structure on M . This metric structure is K-contact if L ξ Φ = 0, that is Φ is K-contact. Henceforth, we will only considercompatible almost contact-complex structures.If ( η, ξ, Φ , g Φ ) is K-contact, because dη ( X, Y ) = 2 g Φ (Φ X, Y ) , we have Φ = ∇ ξ , where ∇ is the Levi-Civita connection of g Φ .If ( η, ξ, Φ , g ) is a contact metric structure, then we define Φ to be normal if(6) N Φ ( X, Y ) = dη ( X, Y ) ⊗ ξ for all X, Y ∈ Γ( T M ) , EFORMATIONS OF CSCS METRICS 5 where N Φ is the Nijenhuis tensor N Φ ( X, Y ) := − Φ [ X, Y ]+Φ([Φ
X, Y ]+[ X, Φ Y ]) − [Φ X, Φ Y ] , for all X, Y ∈ Γ( T M ) . A Sasakian structure ( η, ξ, Φ , g ) can be defined as a normal contact metric structure.Note that (6) implies that L ξ Φ = 0, so it is K-contact.One can alternatively define a Sasakian structure to be a K-contact metricstructure ( η, ξ, Φ , g ) for which the CR structure ( D, J ), where D x := ker η x and J := Φ | D , is integrable, that is(7) N ( D,J ) ≡ , It turns out (see [4]) that (6) is equivalent to L ξ Φ = 0 and (7) for a metric contactstructure ( η, ξ, Φ , g ).We define an almost K-contact-complex structure Φ to be integrable if (7) holds.If Φ is a K-contact almost contact-complex structure compatible with η , then( η, ξ, Φ , g Φ ) is Sasakian if and only if Φ is integrable.One can also define a Sasakian structure, compatible with a Riemannian metricg, to be a unit length Killing field ξ such that the tensor Φ X = ∇ X ξ satisfies thecondition ∇ X Φ( Y ) = g ( ξ, Y ) X − g ( X, Y ) ξ. See [4] for details.We will frequently denote a Sasakian structure by ( g, η, ξ,
Φ) even though spec-ifying ( g, ξ ) , ( g, η ) , or ( η, Φ) is enough to determine the Sasakian structure.2.2.
Transverse K¨ahler structure.
The
Reeb foliation F ξ on M generated bythe action of ξ will be important in the sequel. Note that it has geodesic leaves butin general the leaves are not compact. If the leaves are compact, or equivalently ξ generates an S -action, then ( g, η, ξ, Φ) is said to be a quasi-regular
Sasakianstructure, otherwise it is irregular. If this S action is free, then ( g, η, ξ, Φ) is saidto be regular . In this last case M is an S -bundle over a manifold X , which we willsee below is K¨ahler. If the structure is merely quasi-regular, then the leaf space hasthe structure of a K¨ahler orbifold. In general, in the irregular case, the leaf spaceis not even Hausdorff but we will make use of the transversally K¨ahler property ofthe foliation F ξ which we discuss next.The vector field ξ − √− Iξ = ξ + √− r∂ r is holomorphic on C ( M ). If we denoteby ˜ C ∗ the universal cover of C ∗ , then ξ + √− r∂ r induces a holomorphic actionof ˜ C ∗ on C ( M ). The orbits of ˜ C ∗ intersect M ⊂ C ( M ) in the orbits of the Reebfoliation generated by ξ . We denote the Reeb foliation by F ξ . This gives F ξ atransversely holomorphic structure.The foliation F ξ together with its transverse holomorphic structure is given byan open covering { U α } α ∈ A and submersions(8) Π α : U α → W α ⊂ C m such that when U α ∩ U β = ∅ the mapΦ βα = Π β ◦ Π − α : Π α ( U α ∩ U β ) → Π β ( U α ∩ U β )is a biholomorphism.Note that on U α the differential d Π α : D x → T Π α ( x ) W α at x ∈ U α is an iso-morphism taking the almost complex structure J x to that on T Π α ( x ) W α . Since ξ y dη = 0 the 2-form dη descends to a form ω Tα on W α . Similarly, g T = dη ( · , Φ · )satisfies L ξ g T = 0 and vanishes on vectors tangent to the leaves, so it descends to C. TIPLER AND C. VAN COEVERING an Hermitian metric g Tα on W α with K¨ahler form ω Tα . The K¨ahler metrics { g Tα } and K¨ahler forms { ω Tα } on { W α } by construction are isomorphic on the overlapsΦ βα : Π α ( U α ∩ U β ) → Π β ( U α ∩ U β ) . We will use g T , respectively ω T , to denote both the K¨ahler metric, respectivelyK¨ahler form, on the local charts and the globally defined pull-back on M .If we define ν ( F ξ ) = T M/L ξ to be the normal bundle to the leaves, then we cangeneralize the above concept. Definition 2.2.1.
A tensor Ψ ∈ Γ (cid:0) ( ν ( F ξ ) ∗ ) ⊗ p N ν ( F ξ ) ⊗ q (cid:1) is basic if L V Ψ = 0 for any vector field V ∈ Γ( L ξ ) . Note that it is sufficient to check the above property for V = ξ . Then g T and ω T are such tensors on ν ( F ξ ). We will also make use of the bundle isomorphismΠ : D → ν ( F ξ ), which induces an almost complex structure J on ν ( F ξ ) so that( D, J ) ∼ = ( ν ( F ξ ) , J ) as complex vector bundles. Clearly, J is basic and is mappedto the natural almost complex structure on W α by the local chart d Π α : D x → T Π α ( x ) W α .To work on the K¨ahler leaf space we define the Levi-Civita connection of g T by(9) ∇ TX Y = ( Π ξ ( ∇ X Y ) if X, Y are smooth sections of D, Π ξ ([ V, Y ]) if X = V is a smooth section of L ξ , where Π ξ : T M → D is the orthogonal projection onto D . Then ∇ T is the uniquetorsion free connection on D ∼ = ν ( F ξ ) so that ∇ T g T = 0. Then for X, Y ∈ Γ( T M )and Z ∈ Γ( D ) we have the curvature of the transverse K¨ahler structure(10) R T ( X, Y ) Z = ∇ TX ∇ TY Z − ∇ TY ∇ TX Z − ∇ T [ X,Y ] Z, and similarly we have the transverse Ricci curvature Ric T and scalar curvature s T .We will denote the transverse Ricci form by ρ T . From O’Neill’s tensors computationfor Riemannian submersions [34] and elementary properties of Sasakian structureswe have the following. Proposition 2.2.2.
Let ( M, g, η, ξ, Φ) be a K-contact manifold of dimension n =2 m + 1 , then(i) Ric g ( X, ξ ) = 2 mη ( X ) , for X ∈ Γ( T M ) ,(ii) Ric T ( X, Y ) = Ric g ( X, Y ) + 2 g T ( X, Y ) , for X, Y ∈ Γ( D ) , (iii) s T = s + 2 m. Definition 2.2.3.
A constant scalar curvature Sasakian (cscS) manifold ( M, g, η, ξ, Φ) is a Sasakian manifold with s T constant, or equivalently s g constant. It will be convenient at times to consider the larger class of K-contact structures( g, η, ξ,
Φ) compatible with (
M, η, ξ ). In this case moment map of [10, 23] is thescalar curvature of the
Chern connection ∇ c on ( ν ( F ξ ) , J ) ∇ cX Y = ∇ TX Y − J ∇ TX J ( Y ) . So in considering K-contact structures we will consider a different s Tc than in Propo-sition 2.2.2.Let S ( ξ ) be the space of Sasakian structures (˜ g, ˜ η, ˜ ξ, ˜Φ) on M with ˜ ξ = ξ . For any(˜ g, ˜ η, ˜ ξ, ˜Φ) ∈ S ( ξ ) the 1-form β = ˜ η − η is basic, so [ d ˜ η ] b = [ dη ] b , where [ · ] b denotes EFORMATIONS OF CSCS METRICS 7 the basic cohomology class of a basic closed form. Thus [ ω T ] b ∈ H b ( M/ F ξ , R )(see [4] for more on basic cohomology) is the same for every Sasakian structurein S ( ξ ). Thus, as first observed in [7], fixing the Reeb vector field is the closestanalogue to a polarization in K¨ahler geometry, and we say that the Reeb vectorfield ξ polarizes the Sasakian manifold.We will consider the space of Sasakian structures S ( ξ, J ) with fixed Reeb vectorfield and fixed transversal complex structure J . We define(11) H Φ = { φ ∈ C ∞ b ( M ) : η φ ∧ ( dη φ ) n = 0 } where for any φ ∈ H , we define a new Sasakian structure ( η φ , ξ, Φ φ , g φ ) with thesame Reeb vector field ξ such that(12) η φ = η + d cb φ, Φ φ = Φ − ξ ⊗ d cb φ ◦ Φ , the transversal K¨ahler form is ω Tφ = dη φ = dη + d b d cb φ , and g φ is as in Defini-tion 2.1.3.Note that D and Φ φ vary but ( η φ , ξ, Φ φ , g φ ) has the same transverse holomorphicstructure and same complex structure on C ( M ) as ( η, ξ, Φ , g ) (Prop. 4.1 in [20]).On the other hand, if (˜ η, ξ, ˜Φ , ˜ g ) ∈ S ( ξ, J ) is another Sasakian structure with thesame Reeb vector field ξ and the same transverse complex structure, then thereexists unique functions φ ∈ H , ψ ∈ C ∞ b ( M ) up to addition of a constant and α ∈ H b a harmonic 1-form such that(13) ˜ η = η + α + d cb φ + d b ψ, See [7, Lemma 3.1]. Note that ψ is given by a gauge transformation exp( ψξ ) of M . Since α and d b ψ do not effect the transversal K¨ahler structure they will not beimportant. Thus S ( ξ, ¯ J ) can be viewed as the analogue of the set of K¨ahler metricsin a fixed K¨ahler class.Boyer-Galicki-Simanca [7] proposed to seek the extremal Sasakian metrics torepresent S ( ξ, ¯ J ), by extending Calabi’s extremal problem to Sasakian geometry.We denote by M ( ξ, J ) the metrics associated with Sasakian structures in S ( ξ, J ).We define the Calabi functional by(14) M ( ξ, J ) Cal −→ R g R M ( s − s ) dµ g , where s , the average of s is independent of the structure in S ( ξ, ¯ J ). By Propo-sition 2.2.2.iii s = s T − m where s T = R M s T dµ R M dµ = R M mπc ( F ξ ) ∧ η ∧ (cid:0) ω T (cid:1) m − R M η ∧ (cid:0) ω T (cid:1) m . A Sasaki-extremal metric g ∈ M ( ξ, J ) is a critical point of Cal. As in the K¨ahlercase, the Euler-Lagrange equations of (14) show that this is equivalent to the basicvector field ∂ g s g := ( ∂s g ) being transversely holomorphic. Thus constant scalarcurvature Sasakian metrics are examples, and furthermore a Sasaki-extremal metricis of constant scalar curvature precisely when the transversal Futaki invariant is zero(cf. [7]). The results of this article can be extended to Sasaki-extremal metrics usingrelative K-stability. C. TIPLER AND C. VAN COEVERING
Moment map interpretation.
We will be interested in finding constantscalar curvature Sasakian metrics. Although, much of what follows can be appliedmore generally to Sasaki-extremal metrics by considering that as a relative versionof the cscS case. In [23] W. He gave an interpretation of this problem in term of amoment map, as Donaldson did for the cscK case [10].Let G be the group of strict contactomorphisms . This is the group of diffeomor-phisms f : M → M which satisfy f ∗ η = η . It has Lie algebraLie( G ) = { X ∈ Γ( T M ) , L X η = 0 } , the space of strict contact vector fields. The space of basic functions C ∞ b ( M ) isisomorphic to Lie( G ). For any X ∈ Lie( G ) define H X = η ( X ), and conversely foreach basic function H ∈ C ∞ b ( M ) , there exists a unique strict contact vector field X = X H ∈ Γ( T M ) which satisfies H = η ( X ) , X y dη = − dH. The Poisson bracket is then defined by { F, H } = η ([ X F , X H ]) , and H X H is a Lie algebra isomorphism.We will use the natural G -invariant L inner product on C ∞ b ( M )(15) h f, h i = Z M f h dµ, where dµ = (2 m m !) − η ∧ ( dη ) m is a volume form determined by η .The group of strict contactomorphisms G acts on the space K of K-contactstructures which are compatible with η via( f, Φ) → f ∗ Φ f − ∗ . Moreover, K can be endowed with a K¨ahler structure [23] for which it is aninfinite dimensional symmetric space and for which G acts by biholomorphisms andisometries. First note that T Φ K = { A ∈ End(
T M ) : Aξ = 0 , L ξ A = 0 , A Φ+Φ A = 0 , dη ( AX, Y )+ dη ( X, AY ) = 0 , ∀ X, Y ∈ Γ( T M ) } . An almost-complex structure J is defined on K by J A = Φ A. To each Φ ∈ K , with η we can associate a Sasakian metric g Φ which induces ametric on tensors. We define on K a weak Riemannian metric g K ( A, B ) = Z M h A, B i g Φ dµ η = Z M tr( AB ) dµ η which is Hermitian with respect to J Let Φ be a fixed K-contact structure. Let End S, Φ ( D ) be the basic endomor-phisms of D = ker η symmetric with respect to g T and anti-commuting with Φ .For convenience we identify an endomorphism of D with an endomorphism of T M by acting by zero on the second factor of (5). Define(16) K Φ := { Q ∈ End S, Φ ( D ) : Id − Q > } . EFORMATIONS OF CSCS METRICS 9
We have a chart(17) Ψ Φ : K Φ → K Q Φ ( Id + Q )( Id − Q ) − One can show that Ψ Φ is a bijection. Furthermore, one can easily compute thedifferential d Ψ Φ of Ψ Φ at Qd Ψ Φ : End S, Φ ( D ) → T Φ K A ( Id − Q ) − A ( Id − Q ) − , and check that d Ψ Φ ◦ J = J ◦ d Ψ Φ , where J is the complex structure A Φ ◦ A on K Φ . Therefore the maps (17) are holomorphic charts. Also, arguments as inthe symplectic case show that the 2-formΩ K ( A, B ) = Z M tr(Φ AB ) dµ η , is closed. See [18] and [37] for more details.Define K s Φ , s > n +1 , as in (16) but with sections in Sobolev space L ,s (End S, Φ ( D )),and consider the charts (17) on K s Φ . The above arguments show that the spaceof L ,s K-contact structures K s is a smooth complex Hilbert manifold. And K hasthe structure of a smooth complex ILH-manifold.An almost contact-complex structure Φ can also be identified with a splitting D ⊗ C ∼ = ν ( F ξ ) ⊗ C = T , (Φ ) ⊕ T , (Φ ) , into √− −√− .Suppose Φ is a K-contact complex structure. If Φ is another K-contact complexstructure then Φ = Φ ( Id + Q )( Id − Q ) − , with Q ∈ K Φ . If we extend Q to Q : ν ( F ξ ) ⊗ C → ν ( F ξ ) ⊗ C , then Q = (cid:18) PP (cid:19) , where P : T , (Φ ) → T , (Φ ).This gives a useful complex parametrization of K . Proposition 2.3.1.
Given a K-contact complex structure Φ , the manifold K isparameterized by operators P : T , (Φ ) → T , (Φ ) satisfying the following:(i) After lowering an index P ♭ ∈ Γ( S (Λ , b )) , basic symmetric tensors, and(ii) Id − P P > .And one has T , (Φ) = Im( Id − P ) , T , (Φ) = Im( Id − P ) , where Φ = Φ ( Id + Q )( Id − Q ) − , Q = ( P + P ) . The subspace K i ⊆ K of Sasakian structures is the subvariety for which (7) issatisfied. In the complex parametrization this can be written N ( P ) = ∂ b P + [ P, P ] = 0 . The main result of [23] extends the work of [10] to give the following.
Theorem 2.3.2.
The map µ : K → C ∞ b ( M ) with µ (Φ) = s T (Φ) − s T is an equivariant moment map for the G -action on K , where C ∞ b ( M ) is identified withits dual under the pairing (15).Here s T denotes the Hermitian scalar curvature when Φ is not in K i . The Lichnerowicz operator(18) P Φ : C ∞ b ( M ) → T Φ K , P Φ ( H ) = L X H Φ gives the infinitesimal action of G on K .The theorem reads(19) Ω K ( P Φ ( H ) , A ) = h H, Ds T ( A ) i , ∀ A ∈ T Φ K . If we were in the finite dimensional situation, then the Kempf-Ness theoremwould lead to the identification K s // G C = µ − (0) / G , where K s are the polystable points in K and G C is the complexified group. Aconstant transverse scalar curvature metric, which is a zero point of the momentmap µ , would correspond to a polystable complex orbit of the G C action.But there are two problems in this situation. First, in this infinite dimensionalsituation there is no local compactness allowing the usual arguments. And secondly,the complexification G C of G does not exist as a group.Although G C does not exist we can define the action of the complexified Liealgebra C ∞ b ( M, C ) of G on K since it is a complex manifold. We extend (18) to(20) P Φ : C ∞ b ( M, C ) → T Φ K , by taking √− H , H ∈ C ∞ b ( M ) to Φ L X H Φ. Then we say that a smooth pathΦ( t ) ∈ K lies in an orbit of G C if(21) ˙Φ( t ) ∈ Im P Φ( t ) , ∀ t. Note that the integrability of Φ ∈ K i does not imply that L Φ X φ Φ equals Φ L X φ Φ.In fact, an easy computation using (6) shows that(22) L Φ X φ Φ( Y ) = Φ L X φ Φ( Y ) + dφ ( Y ) ⊗ ξ, for Y ∈ T M.
But this shows one does have equality on the level of the transverse K¨ahler structure( ω T , J ), since the last term acts trivially on basic forms. Proposition 2.3.3.
Let ( M, η, ξ, Φ , g ) be Sasakian. Then up to a diffeomorphismpreserving the Reeb foliation F ξ the G C orbit of Φ consists precisely of all structures ( η φ , ξ, Φ φ , g φ ) with φ ∈ C ∞ b ( M ) as in (12).Proof. Let φ ∈ H Φ . Define the Sasakian structure ( η t , ξ, Φ t ) , ≤ t ≤ , (23) η t = η + td c φ, Φ t = Φ − tξ ⊗ dφ. Let X tφ be the strict contact vector field for η t with Hamiltonian φ , and define V t = Φ t X tφ . We have L V t η t = V t y dη t = − d c φ. If f t is the flow associated to V t , then ddt f ∗ t η t = f ∗ t L V t η t + f ∗ t d c φ = 0 . EFORMATIONS OF CSCS METRICS 11
Therefore f ∗ t η t = η , and the structure ( η t , Φ t ) is isometric to ( η, f ∗ t Φ t ), where f ∗ t Φ t = f − t ∗ ◦ Φ t ◦ f t ∗ .We claim that ( η, f ∗ t Φ t ) is in the orbit of G C . In fact ddt f ∗ t Φ t = f ∗ t L V t Φ t + f ∗ t (cid:0) − ξ ⊗ dφ (cid:1) = f ∗ t (cid:0) Φ t L X tφ Φ t (cid:1) = (cid:0) f ∗ t Φ t (cid:1) L X f ∗ t φ (cid:0) f ∗ t Φ t (cid:1) , (24)where X f ∗ t φ is the contact vector field for η with Hamiltonian f ∗ t φ . Thus f ∗ t Φ t satisfies (21).Conversely, suppose that Φ t is a smooth path of almost contact-complex struc-tures so that ( η, ξ, Φ t ) is Sasakian and Φ = Φ. Suppose this is contained in the G C orbit of Φ. After possibly acting by contactomorphisms we may assume(25) ddt Φ t = Φ t L X φt Φ t , where φ t ∈ C ∞ b ( M ) is a smooth path and X φ t is the associated contact vector field.Let f t be the flow of Φ t X φ t . Then ddt f ∗ t η = f ∗ t (cid:0) − d ct φ t (cid:1) = − d c (cid:0) f ∗ t φ t (cid:1) , where d ct is with respect to the transversal complex structure induced by Φ t . Thesecond equality is because f ∗ t Φ t and Φ induce the same transversal complex struc-ture from (22). And(26) f ∗ t η − η = − d c Z t f ∗ s φ s ds. Define η t = f ∗ t η , then ( η, Φ t ) is isometric to ( η t , f ∗ t Φ t ). We have η t = η + d c H t where H t = − R t f ∗ s φ s ds . But since f ∗ t Φ t and Φ induce the same transversalcomplex structure, we must have f ∗ t Φ t = Φ − ξ ⊗ dH t . (cid:3) Remark 2.3.4.
This orbit does not effect the harmonic H b or exact componentsin (13). The exact component in (13) is given by the diffeomorphism exp( ψξ ) Thus S ( ξ, J ) consists of the G C orbit of φ and variations by H b . But since these variationsdo not affect the transversal K¨ahler structure, this wont cause any issues.2.4. K-stability for Sasakian manifolds.
T. Collins and G. Sz´ekelyhidi [42]defined test configurations and K-polystabiliy for irregular Sasakian manifolds ex-tending the work of J. Ross and R. Thomas [36] on the quasi-regular case. Todefine test configurations for Sasakian manifolds, Collins and Sz´ekelyhidi gave analgebraic interpretation of Reeb vector fields on Sasakian manifolds which we recallnow. Let (
M, g, η, ξ,
Φ) be a Sasakian manifold, then the metric cone ( C ( M ) = R + × M, g = dr + r g ) over ( M, g ) is K¨ahler for the complex structure I . Asexplained in [42, Section 2], the cone Y = C ( M ) ∪ { } is an affine variety withisolated singularity at 0.We may consider the cone ( Y, ξ ) polarized by the Reeb vector field representing apolarized Sasakian manifold. This is because any two Sasakian structures ( g, η, ξ,
Φ)and (˜ g, ˜ η, ξ, ˜Φ) with the same polarized cone ( Y, ξ ) differ as in (13) with α = 0. The Reeb vector field generates a torus action T ⊂ Aut( Y ) with ξ ∈ t = Lie( T ), whichof course extends to an algebraic torus action T ⊂ Aut( Y ). Let O Y be the structuresheaf of Y and consider the weight decomposition H ( Y, O Y ) = X α ∈W T H ( Y, O Y ) α where W T ∼ = Z k , k = dim C T , are the weights of the T -action.Then from η ( ξ ) > α ( ξ ) > α ∈W T \ { } . It turns out that the fact that the Reeb vector field acts with positiveweights on the non-constant functions of Y gives an algebraic characterization ofthe Reeb cone: { ξ ′ ∈ t : η ( ξ ′ ) > } = { ξ ′ ∈ t : −√− α ( ξ ′ ) > , ∀ α ∈ W T \ { }} . It suggests the following algebraic definition of a Reeb field:
Definition 2.4.1.
A Reeb field on an affine scheme Y with torus T ⊂ Aut Y is anelement ξ ′ ∈ t such that −√− α ( ξ ′ ) > ∀ α ∈ W T \ { } . On the polarized cone (
Y, ξ ) it remains to define the notion of a compatibleK¨ahler metric.
Definition 2.4.2.
A K¨ahler metric on an affine scheme Y is compatible with a Reebfield ξ ∈ t if there exists a ξ -invariant function r : Y → R + such that ω = √− ∂∂r and ξ = I ( r∂ r ) where I is the almost complex structure on Y . Any polarized affine variety (
Y, ξ ) smooth except at possibly one point admits aK¨ahler metric ω compatible with the Reeb field ξ , and ( Y, ξ, ω ) is the metric coneover a Sasakian manifold. To see this let T be the torus generated by ξ and choosesufficiently many T -homogenous generators { f , . . . , f d } , f i ∈ H ( Y, O Y ) α i , α i ∈W T , then ( f , . . . , f d ) : Y → C d is an embedding with T acting diagonally on C d . Then there exists a Sasakianstructure on the sphere S d − with Reeb vector field ˆ ξ restricting to ξ . In particu-lar, a Sasakian manifold ( M, g, η, ξ,
Φ) can equivalently be defined as an algebraicscheme Y = C ( M ) ∪ { } smooth away from 0 with Reeb vector field ξ and com-patible K¨ahler metric ω .We can now recall the definition of test configurations for Sasakian manifoldsfrom [42]. Let Y be an affine variety polarized by a Reeb vector field ξ ∈ t , with t = Lie( T ) and T ⊂ Aut( Y ). Definition 2.4.3.
A T-equivariant test configuration for Y is a set of T -homogeneouselements { f , ..., f k } that generate H ( Y, O Y ) in sufficiently high degrees togetherwith a set of integers { w , ..., w k } . This definition generalizes the usual test configurations for polarized manifoldsor orbifolds. Given the set of generators { f j } , we can embed Y into C k and considerthe C ∗ -action on C k with weights ( w , . . . , w k ). Then the flat limit Y over 0 ∈ C of the C ∗ -orbit of Y provides a flat family of affine schemes over C . Moreover, thecentral fiber Y is invariant under the C ∗ action defined by the weights { w j } . We EFORMATIONS OF CSCS METRICS 13 will say that this test configuration is a product configuration if Y is isomorphic to Y . Let υ be a generator of the C ∗ action on Y defined by the weights { w j } . Aweight is assigned to the test configuration, Fut( Y , ξ, υ ), the so-called Donaldson-Futaki invariant. This weight was first defined by Futaki in the smooth case [19],and then generalized by Donaldson to the algebraic setting [11]. Lastly, making useof the Hilbert series, Collins and Sz´ekelyhidi managed to extend the definition tothe Sasakian case. See [42, Definition 5.2.]. Definition 2.4.4.
A polarized affine variety ( Y, ξ ) is K-semistable if, for everytorus T , ξ ∈ Lie( T ) , and every T -equivariant test configuration with central fiber Y , the Donaldson-Futaki invariant satisfies Fut( Y , ξ, υ ) ≤ with υ a generator of the induced C ∗ action on the central fiber.It is K-polystable if the equality holds if and only if the T -equivariant test con-figuration is a product configuration. We will only make use of the Futaki invariant for smooth test configurations,so we wont need it in its full generality. In [7] the Futaki invariant is adapted tothe Sasakian case. In this case it gives a character on the transversely holomorphicvector fields. A transversely holomorphic vector field is a complex vector field X which projects to a holomorphic vector field on every holomorphic foliationchart (8). We assume that X is Hamiltonian, i.e. has a potential, so there is an H ∈ C ∞ b ( M, C ) with ∂H = − dη ( X, · ). Then the Futaki invariant is(27) F ξ, Φ ( X ) = − Z M H ( s − s ) dµ η , and only depends on the polarization and transversal complex structure.The next lemma essentially shows that for smooth Sasakian manifolds both thesedefinitions are the same. Lemma 2.4.5.
Let ( M , ξ , Φ ) be a polarized Sasakian manifold with correspond-ing affine K¨ahler cone ( Y = C ( M ) , ξ, ω ) . Suppose υ is the generator of a holo-morphic C ∗ -action on Y commuting with ξ . Then, there is a constant c ( n ) > depending only on the dimension such that F ut ( Y , ξ, υ ) = c ( n ) F ξ, Φ ( υ M ) . Proof.
Since υ | M commutes with ξ , it induces a transversally holomorphic vectorfield υ M on ( M , ξ, Φ ).First, suppose ξ is quasi-regular. We will show the Donaldson-Futaki invariantcomputed on the quotient polarized orbifold ( X, L ) is equal to (27). We basicallyextend the computations in [36, Sect. 2.9]. Recall from [36] that the orbifoldRiemann-Roch gives h ( X, L k ) = a k n + a k n − + ˜ o ( k n − )(28) w ( H ( X, L k )) = b k n +1 + b k n + ˜ o ( k n ) , (29)where w ( H ( X, L k ) is the total weight of υ . Here ˜ o ( k n − ) means a sum of terms in k lower order than n − r ( k ) δ ( k ) where r ( k ) is a polynomialof degree k and δ ( k ) is periodic in k of period Ord( X ) and average 0. Let ω ∈ c ( L ) be the orbifold K¨ahler form on X . Using the induced metric on K orbX we have the Ricci form ρ ∈ − πc ( K orbX ). The coefficients a = 1 n ! Z X ω n , a = 14 π ( n − Z X ω n − ∧ ρ and b = 1 n ! Z X Hω n , were computed in [36], where H is the Hamiltonian of υ .Let O P (1) ∗ be the principal C ∗ -bundle associated to O (1). Form the associated( X, L )-bundle(30) ( X , L ) := O P (1) ∗ × C ∗ ( X, L ) . If π : X → P is the projection then π ∗ L k is the associated bundle of the C ∗ -representation H ( X, L k ). We have (cf. [36])(31) w ( H ( X, L k )) = χ ( X , L ) − χ ( X, L k ) , where for k >> b = − n ! Z X c ( L ) n c ( K orb X ) − n ! Z X c ( L ) n . It was shown by Ross and Thomas (see also [12]) that for associated bundle in(30) c ( L ) = Hω F S + ω , where ω F S is the Fubini-Study metric on P . Note that K orb X = π ∗ K P ⊗ K , where K is the associated bundle on X to K X as in (30). Thenthe same argument gives c ( K orb X ) = ( f − ω F S − π ρ, where f is the “Hamiltonian” for the action of υ on K X . Substituting these into(32) give b = − n ! Z X ( ω n + nHω F S ∧ ω n − ) ∧ (( f − ω F S − π ρ ) − n ! Z X ω n = − n ! Z X ( f − ω F S ∧ ω n − n π Hω F S ∧ ω n − ∧ ρ − n ! Z X ω n = − n ! Z X f ω n + 14 πn ! Z X sHω n = 14 πn ! Z X sHω n , where the last step follows because f can be shown to be the divergence of υ .Recall that (cf. [36] and [42]Fut( Y , ξ, υ ) = a b − a b a , then substituting the expressions for a , a , b and b givesFut( Y , ξ, υ ) = 14 πn ! Z X H ( s − s ) ω n . For the general case first note that if we rescale ˜ ξ = cξ, c > , thenFut( Y , ˜ ξ, υ ) = c − ( n +1) Fut( Y , ξ, υ ) . EFORMATIONS OF CSCS METRICS 15
And similarly F ˜ ξ, Φ ( υ ) = c − ( n +1) F ξ, Φ ( υ ) . So the lemma is proved for any ξ proportional to an integral element of t = Lie( T ).In the irregular case, by [42, Corollary 2], there is a sequence ξ j of Reeb vectorfields proportional to integral vector fields on Y such that ξ j → ξ ∈ t . Both thetransversal Futaki invariant and the Donaldson-Futaki invariant depend continu-ously on ξ j and the result follows at the limit. (cid:3) Deformations and stability
We are interested in the deformation theory of cscS Sasakian metrics. In thissection we show how the relationship between cscS metrics and stability in theGIT sense can be used to give an algebraic criterion for deformation of canonicalSasakian metrics.3.1.
Deformation complexes.
We describe two complexes relevant to the de-formations of a Sasakian structure (
M, η, ξ, Φ ) that we will consider. The firstdescribes deformations of the transversal complex structure of the Reeb foliation( F ξ , J ).Define A k := Γ(Λ ,kb ⊗ ν ( F ) , ). We have the complex(33) 0 → A ∂ b −→ A ∂ b −→ A → · · · , which we denote by A • . This is the basic version of the complex used by Kuran-ishi [27] whose degree one cohomology H ( A • ) is the space of first order deforma-tion of the transversal complex structure J modulo foliate diffeomorphisms. In [16]and [22] it was shown that H ( A • ) is the tangent space to a versal deformationspace of ( F ξ , J ) preserving the smooth foliation F ξ .For the second complex, let E k , k ≥ , be the kernel of the mapΛ ,kb ⊗ ν ( F ) , ∼ = Λ ,kb ⊗ Λ , → Λ ,k +1 b . Define B k := Γ( E k ) , k ≥
1, and B := C ∞ b ( M, C ). Note that B = T Φ K , and wedefine a complex B • by(34) 0 → C ∞ b ( M, C ) P −→ T Φ K ∂ b −→ B → · · · , where P is (20). The remaining maps are the same operators ∂ b as above. Then H ( B • ) is the space of first order deformations of Φ modulo the action of G C .There is a mapping of the complex B • to A • . In degree zero, this is C ∞ b ( M, C ) −→ A f ( ∂f ) ♯ where ( · ) ♯ denotes index raising by the transversal K¨ahler metric. And for k ≥ Proposition 3.1.1.
The induced map in cohomology H k ( B • ) → H k ( A • ) , k ≥ ,is injective if H ,kb = 0 and is surjective if H ,k +1 b = 0 , where H , • b denotes thetransversal Dolbeault cohomology. Proof.
Suppose H ,kb = 0 and β ∈ B k , k ≥
2. If [ β ] = 0 in H k ( A ), there existsa γ ∈ A k − with ∂ b γ = β . Since ∂ b γ ∈ B k , ∂ skw( γ ♭ ) = skw( ∂ b γ ♭ ) = 0, where ∂ is the ordinary Dolbeault operator and γ ♭ is the section of Λ ,k − b ⊗ Λ , b obtainedfrom the transversal K¨ahler form. By assumption there is an α ∈ Γ(Λ ,k − b ) with ∂α = skw( γ ). Let θ = γ − ∂ b α ♯ . Since skw( ∂ b α ♯ ) = ∂α = skw( γ ♭ ), θ ∈ B k − . Since ∂ b θ = β , [ β ] = 0 in H k ( B ).If k = 1 and there exists a γ ∈ A with ∂ b γ = β , then ∂γ ♭ = 0. There exists an f ∈ C ∞ b ( M, C ) with ∂f = γ ♭ . Thus P f = ∂ b ∂ ♯ f = β .Suppose β ∈ A k , k ≥ , ∂ b β = 0 and H ,kb = 0. Write β = β + β with respectto A k = B k ⊕ λ ,k +1 b . Then 0 = skw( ∂ b β ♭ ) = skw( ∂ b β ) = ∂β , and there exists γ ∈ Γ(Λ ,kb ) with ∂γ = β . One easily sees that β − ∂ b γ ♯ ∈ B k . (cid:3) Construction of the slice.
We will construct a slice for the complex (34)on a Sasakian manifold (
M, η, ξ, Φ ). The transverse metric and the L inner prod-uct on forms enable us to define Sobolev norms on B • and we can define adjointoperators P ∗ and ∂ ∗ b . Then the space H ( B • ) ≃ ker (( ∂ ∗ ∂ ) + PP ∗ ) encodes in-finitesimal deformations of the transverse complex structures that are compatiblewith η modulo the action of G C . This space is finite dimensional as it is the kernelof the fourth order transversely elliptic operator (cid:3) Φ = ( ∂ ∗ b ∂ b ) + PP ∗ (cf. [17]).Let G be the stabilizer of Φ in G , then G = Aut( M, η, ξ, Φ ) and is thus com-pact. This group acts linearly on T Φ K and on H ( B • ). The group G also has acomplexification G C which also acts on these spaces. Proposition 3.2.1.
There is a holomorphic G -equivariant map S from a neigh-borhood of zero B in H ( B • ) into a neighborhood of Φ in K such that the G C orbitof every integrable Φ close to Φ intersects the image of S . Moreover, if x and x ′ lie in the same G C orbit in U and S ( x ) ∈ K i , then S ( x ) and S ( x ′ ) are in the same G C orbit in K .Proof. This follows from arguments of M. Kuranishi [27] adapted to the transver-sally complex situation and with the complex B • replacing A • .Recall that K Φ ⊂ T Φ K = B is an open subset. Let U := K Φ ∩ ker (cid:3) Φ .Then the restriction of the map (17) to U gives holomorphic map Ψ : U → K . Weparametrize complex structures by maps P : T , (Φ ) → T , (Φ ), i.e. P ∈ B , asin Proposition 2.3.1.Then as in [27] we construct an injective holomorphic map ϑ : B → B on a G invariant restriction B ⊂ U so that ϕ = ϑ ( P ) is in K i and satisfies N ( ϕ ) = ∂ b ϕ + [ ϕ, ϕ ] = 0 and P ∗ ϕ = 0precisely when H (cid:0) [ ϑ ( P ) , ϑ ( P )] (cid:1) = 0 , where H is the projection onto the harmonic space. We define the slice to be S = Ψ ◦ ϑ : B → K . Thus the structures in K i are parametrized by the vanishing EFORMATIONS OF CSCS METRICS 17 of a holomorphic map Θ : B → H ( B • ),Θ( P ) = H (cid:0) [ ϑ ( P ) , ϑ ( P )])The remaining properties follow from the arguments in [9, Lemma 6.1] adaptedto the transversally holomorphic situation. (cid:3) Remark 3.2.2.
The slice S of the proposition is holomorphic with B and K giventhe L ,ℓ topology, i.e. of Banach manifolds. We will fix a slice for some large ℓ .3.3. Reduction to finite dimensional GIT.
Assume now that (
M, η, ξ, Φ ) isa cscS manifold. We consider the problem of finding cscS structures on nearby( M, η, ξ,
Φ). In order to get a finite dimensional moment map we will perturb theholomorphic slice in Proposition 3.2.1.
Proposition 3.3.1.
There exists a G -equivariant C map ˆ S from a neighborhood B of in H ( B • ) to K with ˆ S (0) = Φ , such that µ ◦ ˆ S = ( s T − s T ) ◦ ˆ S takes valuein Lie( G ) .Furthermore, the G C orbit of every integrable smooth Φ close to Φ intersects theimage of ˆ S . If x and x ′ lie in the same G C orbit in B and ˆ S ( x ) ∈ K i , then ˆ S ( x ) and ˆ S ( x ′ ) are in the same G C orbit in K . Moreover, ˆ S is tangent to S at 0 to firstorder.Here we take C to mean that ˆ S is C as a map into K ℓ , the space of L ,ℓ K-contact structures, for some large ℓ . It follows that ˆ S is C as a map from B intothe space of C m K-contact structures, for ℓ > n + m .Proof. We will perturb the map S of Proposition 3.2.1 along G C -orbits to obtainour new map ˆ S with the stated properties. Let B ⊂ H ( B • ) be a G -invariantneighborhood of zero. We will identify Lie( G ) with the corresponding subspace ofbasics functions. Let W ,kb be the orthogonal complement of Lie( G ) in L ,kb ( M, R ),Π k : L ,kb ( M, R ) → W ,kb the orthogonal projection on this space, and U k a smallneighborhood of zero inside W ,kb , where we assume k > n + 2. Given φ ∈ U k and K-contact structure Φ with values in L ,ℓb , for large ℓ , we define F (Φ , φ ) tobe the structure f ∗ ( η , Φ ), where ( η , Φ ) = ( η + d c φ, Φ − ξ ⊗ dφ ) and f is thediffeomorphism as in Proposition 2.3.3. We would like to apply the implicit functiontheorem to(35) H : B × U k → U k − ( x, φ ) Π k − ( s T ( F ( S ( x ) , φ ))) , which would give the desired perturbed section ˆ S . But it is not clear that H isdifferentiable, since f is a L ,k − diffeomorphism so F ( S ( x ) , φ ) has only L ,k − regularity. So we must employ a slightly more complicated argument.Let D be the group of diffeomorphism of M . This is an ILH Lie group [33, 14],which is the inverse limit of D k , k > n + 2 , the group of L ,k diffeomorphisms of M . We define χ i : B × U k → D k − − i , where χ i ( x, φ ) is the diffeomorphism f of Proposition 2.3.3 depending on theSasakian structure ( η + d c φ, S ( x ) − ξ ⊗ dφ ). It is known, see [15], that χ i is a C i map. In particular, χ := χ is C . Also, we have ψ : B × U k → D k − , defined by ψ ( x, φ ) := χ ( x, φ ) − which can be seen to be C as follows. Recall that f is the t = 1 diffeomorphism of the flow of V t = Φ t X tφ as defined in Proposition 2.3.3.Then ψ ( x, φ ) is the t = 1 diffeomorphism of the flow of − V − t , thus ψ is C .Let Lie( G ) have an orthonormal basis { σ , . . . , σ r } ⊂ C ∞ b . Then defineΠ k = Id − Π Gk , where Π Gk : L ,kb → Lie( G ) is the orthogonal projection,Π Gk ( f ) = r X j =1 σ j h f, σ j i L . Instead of H we consider(36) F : B × U k → U k − ( x, φ ) Π k − (cid:0) ψ ( x, φ ) ∗ Π k − (cid:1) s T ( S ( x ) , φ ) , where s T ( S ( x ) , φ ) is the transverse scalar curvature of ( η + d c φ, S ( x ) − ξ ⊗ dφ ). Wehave ψ ( x, φ ) ∗ Π k − = Id − ψ ( x, φ ) ∗ Π Gk − and ψ ( x, φ ) ∗ Π Gk − ( f ) = ψ ( x, φ ) ∗ ◦ Π Gk − ◦ χ ( x, φ ) ∗ f = ψ ( x, φ ) ∗ (cid:0) r X j =1 σ j h χ ( x, φ ) ∗ f, σ j i (cid:1) = r X j =1 ψ ( x, φ ) ∗ σ j h f, ψ ( x, φ ) ∗ σ j i η + d c φ , is easily seen to be C on B × U k . Because σ j ∈ C ∞ b we have that ψ ( x, φ ) ∗ σ j is a C mapping from U × U k to L ,k − b (cf. [15]). Therefore F is C .The derivative of F with respect to U k at the origin is DF (0 , = P ∗ P : W ,k → W ,k − , which is an isomorphism for all k > n + 2. Applying the implicit function theoremfor k ≫
1, we get a C mapping U ∋ x φ ( x ) ∈ L ,kb ( M ) , so that F ( x, φ ( x )) = 0.After possibly shrinking B this implies that on B (37) (cid:0) ψ ( x, φ ( x )) ∗ Π k − (cid:1) s T ( S ( x ) , φ ( x )) = 0 . Since φ ( x ) ∈ L ,kb ( M ) for each x ∈ B , we have ψ ( x, φ ( x )) ∈ D k − for all x ∈ B .From (37) we have s T ( S ( x ) , φ ( x )) ∈ Im h ψ ( x, φ ( x )) ∗ Π G i , where the right hand side is a finite dimensional spaces of L ,k − function. Notethat on any local transverse chart of the Reeb foliation F ξ as in (8) the map φ s T ( S ( x ) , φ ) is elliptic. By well known elliptic regularity φ ( x ) ∈ L ,k +2 b ( M ).But then s T ( S ( x ) , φ ( x )) ∈ L ,kb ( M ), so elliptic regularity implies φ ( x ) ∈ L ,k +4 b ( M ).Continuing, we see that φ ( x ) ∈ C ∞ b ( M ) for each x ∈ B .It follows that(38) ˆ S ( x ) = ( η, Φ( x )) := χ ( x, φ ( x )) ∗ (cid:0) η + d c φ ( x ) , S ( x ) − ξ ⊗ dφ ( x ) (cid:1) EFORMATIONS OF CSCS METRICS 19 is section of K , which is C as a map from B to K ℓ for ℓ = k −
5. Note that at thecost of shrinking U , one can choose ℓ arbitrarily large. This is also a C map from U to C mb ( M ) for ℓ > n + m by the Sobolev embedding theorem.The statements about the orbits of G C follow from the same properties of theslice S of Proposition 3.2.1 since ˆ S is constructed by deforming S along G C orbits.The G -equivariance of ˆ S can be easily checked from that of S and the othermaps involved. Suppose A ∈ T Φ K is tangent to ˆ S . We have h ds T ( A ) , H i = g K (Φ P Φ ( H ) , A ) = 0 , for all H ∈ C ℓb , since ds T ( A ) ∈ Lie( G ) and P Φ ( H ) = 0 for H ∈ Lie( G ). Thus A is orthogonalto Im(Φ P Φ ) and one can check by differentiating (38) that the component ofIm d ˆ S (0) not tangent to S is in Im(Φ P Φ ). (cid:3) Let U ⊂ H ( B • ) be a G-invariant neighborhood as above, then the pullback ofthe moment map µ = s T − s T to U by ˆ S is a moment map for the G -action on U with respect to the pullback of the symplectic form Ω K on K by ˆ S , denoted Ω. Notethat the pulled back moment map µ on U is C and Ω is C . We have reduced theproblem of finding zeros of the moment map µ to a finite dimensional Hamiltoniansystem ( U, G, Ω , µ ). But note that this problem is slightly complicated by the factthat Ω is not K¨ahler. Proposition 3.3.2.
Suppose that x ∈ U , after possibly shrinking U , is polystablefor the G C -action on H ( B • ) . Then there is y in the G C -orbit of x such that s T ( ˆ S ( y )) − s T = 0 . If in addition ˆ S ( x ) is integrable, then the corresponding cscSmanifold ( M, η, ˆ S ( y )) is a deformation of ( M, η, ˆ S ( x )) as in (12).Proof. Let Ω be the restriction of the symplectic form Ω K of K to U ⊂ H ( B • )via ˆ S . Thus the restriction of the moment map µ : U → Lie( G ) is a moment mapfor G . Also, let Ω be the linear symplectic form on H ( B • ) induced by Ω at theorigin. And write ν : H ( B • ) → Lie( G )for the corresponding moment map for the flat K¨ahler structure (Ω , J ), where J is the vector space complex structure on H ( B • ). Let σ x : Lie( G ) → T x U be theinfinitesimal action, then h ν ( x ) , ξ i = 12 Ω ( σ x ξ, x ) , for ξ ∈ Lie( G ) . We have µ (0) = 0 by assumption, and dµ = 0 because 0 ∈ U is a fixed point ofthe G -action. It is also routine to check that d dt µ ( tx ) | t =0 = 2 ν ( x ) . Therefore by Taylor’s theorem we have(39) µ ( tx ) − ν ( tx ) = o ( t ) , with lim t → | µ ( tx ) − ν ( tx ) | t = 0 converging to 0 uniformly in x ∈ U . Given δ > C > U so that Ct − | µ ( tx ) − ν ( tx ) | < δ .If x ∈ U is polystable for the G C -action, then by the Kempf-Ness theorem thereis a zero x of ν in the G C orbit of x . Since this is given by minimizing the normover G C · x , the zero will be in U . For x ∈ U ⊂ H ( B • ) let K x ⊂ G the stabilizer of x and k x its Lie algebra. Wehave K tx = K x . Then for ξ ∈ k x ddt h µ ( tx ) , ξ i = Ω tx ( σ tx ( ξ ) , x ) = 0 . Thus for all x ∈ U we have µ ( x ) ∈ k ⊥ x .We also shrink U so that Ω( X, JX ) ≥ Ω ( X, JX ) , ∀ X ∈ T U .Define functions h = k ν k and f = k µ k on U , where the norms are givenby invariant metrics on Lie( G ). Note that f is the Calabi functional. We have dh x ( w ) = 2Ω ( σ x ν ( x ) , w ) , w ∈ T x U .Let x be a point with ν ( x ) = 0. We consider the restriction of f and h to theorbit G C · x . Note that both f and h are invariant under G . Let R ⊂ G C · x bea local slice through x for the G action on G C · x invariant under K x . Clearly dh x = 0. The Hessian at of h at x is(40) d x h ( v, w ) = 2 m X j =1 Ω ( σ x e j , v )Ω ( σ x e j , w ) , where { e , . . . , e m } is an orthonormal basis of k ⊥ x . If v ∈ T x R then v = Jσ x η for η ∈ k ⊥ x . And d x h ( v, v ) = m X j =1 g ( σ x e j , σ x η ) > , for v = 0.There exists constants ǫ > δ > δ > h < ǫ on B δ ( x ), ballof radius δ in R , while h > ǫ on A = B δ ( x ) \ B δ ( x ). We will identify theslice R and the above neighborhood in the obvious way when x is replaced with tx , < t <
1. By shrinking U we may assume k µ ( tx ) − ν ( tx ) k ≤ ǫt , whichimplies that k µ ( tx ) k ≥ ǫt on A and k µ ( tx ) k ≤ ǫt on B δ . So on the slice R through tx there exists a minimum of f at x ∈ B δ . Since0 = df x ( v ) = 2Ω( σ x µ ( x ) , v ) on G C · tx we have0 = 2Ω( σ x µ ( x ) , Jσ x µ ( x )) ≥ Ω ( σ x µ ( x ) , Jσ x µ ( x ))= g ( σ x µ ( x ) , σ x µ ( x )) . Thus µ ( x ) ∈ k x and µ ( x ) = 0 since we have µ ( x ) ∈ k ⊥ x . (cid:3) This proposition gives an algebraic, finite dimensional way to obtain deforma-tions of cscS metrics.
Remark 3.3.3.
In the K¨ahler case, i.e. (
M, η, ξ, Φ ) is a regular Sasakian manifold,gives the local moduli of csc metrics. The proposition proves that locally everypolystable orbit contains a csc metric. The results of X. Chen and S. Sun [9] showthe necessity and uniqueness. More precisely, they show that only polystable orbitscontain csc metrics and they are unique up to the action of G . Thus the local EFORMATIONS OF CSCS METRICS 21 moduli is given by a neighborhood of zero in the GIT quotient H ( B • ) // G C . Thenecessity of polystability and uniqueness in the csc Sasakian case is still open. Remark 3.3.4.
All these results extends to the extremal setting, using relativeversions of group actions containing the extremal vector field. See [35].3.4.
Deformations and K-stability.
In this section, we show that if a Sasakianmanifold is obtained by a small deformation of the transverse complex structureof a cscS manifold, then K -polystability is a sufficient condition to admit a cscSmetric. We also show that a small deformation of a cscS metric is K-semistable. Theorem 3.4.1.
Let ( M, η, ξ, Φ ) be a cscS manifold and ( M, η, ξ, Φ) a nearbySasakian manifold with transverse complex structure ¯ J . Then if ( M, η, ξ, Φ) isK-polystable, there is a constant scalar curvature Sasakian structure in the space S ( ξ, ¯ J ) .Proof. Consider the map ˆ S from Proposition 3.3.1. If Φ is close enough to Φ , thenup to the action of G C there is x ∈ U ⊂ H ( B • ) such that ˆ S ( x ) = Φ. Assumefor the moment that the K-polystability of ( M, η,
Φ) implies the polystability of x under the action of G C in U , where G is the stabilizer of Φ in K under the G action. Then by proposition 3.3.2 the result follows. What remain to be shown isthat if x is not polystable in U , then ( M, η,
Φ) is not K-polystable.Consider a one parameter subgroup ρ : C ∗ → G C such that ∃ lim λ → ρ ( λ ) · x = x ′ ∈ U with x ′ polystable. Note that we can assume ρ ( S ) ⊂ G , the stabilizer of Φ . Wewill build a destabilizing test configuration for ( M, η,
Φ) from this one parametersubgroup.Note that the polarized affine cones defined by (
M, ξ, ˆ S ( x )) and ( M, ξ, S ( x )) arebiholomorphic. In fact by the construction of ˆ S the Sasakian structures differ asin Proposition 2.3.3. So it is enough to construct a destabilizing test configurationfor ( M, ξ, S ( x )).We have a holomorphic map F : ∆ → K i , where ∆ ⊂ C is the unit disk, F ( λ ) = S ( ρ ( λ ) · x ) for λ ∈ ∆ \ { } , and F (0) = S ( x ′ ).Our test configuration will be Y = Y × ∆ as a smooth manifold with Y , the conefor S ( x ). The complex structure on Y × { t } is given by F ( t ), and the rest of thecomplex structure is given by the holomorphic map F . The S -action on Y is givenby ρ ( τ )( y, t ) = ( ρ ( τ ) y, τ t ) , and extends to a holomorphic C ∗ -action. Furthermore, the real torus T generatedby ξ acts fiber-wise on Y .To see this gives a test configuration as in Definition 2.4.3 choose a subgroup S ⊂ T generated by ζ in the Reeb cone. Then quotienting by the C ∗ generatedby this action gives an orbifold test configuration ( X , L ) where L is a positiveorbifold bundle, with the C ∗ -action generated by ρ . As in [36] we can embed X in a weighted projective bundle P ( ⊕ i ( π ∗ L w i ) ∗ ) over ∆, for large enough w i ∈ N , wherethe summands are preserved by T . Then the ρ action on ( X , L ) induces a C ∗ -action on V = ⊕ i H ( X , L w i ). This induces a diagonal action on V × C inducing ρ on Y ⊂ V × C . Then the { f i } come from picking T -homogeneous elements ofthe summands of V , and the { w i } are the weights of ρ on these elements. Thus wehave T -equivariant test configuration Y for ( Y , ξ ) with central fiber Y .Since x ′ is polystable, Y = ˆ S ( x ′ ) has a cscS structure in its G G orbit fromProposition 3.3.2. From Lemma 2.4.5, the Donaldson-Futaki invariant of this testconfiguration is the usual transversal Futaki invariant which vanishes because Y admits a cscS structure. Moreover, the stabilizer of Φ ′ = S ( x ′ ) in G is strictlygreater than that of Φ because x ′ is in the closure of the orbit of x . Then theautomorphism group of Y is smaller than the automorphism group of Y and thistest configuration is not a product. Thus ( M, η,
Φ) is not K -polystable, which endsthe proof. (cid:3) Theorem 3.4.2.
Let ( M, η, ξ, Φ ) be a cscS manifold. Then any small deformation ( M, η, ξ, Φ) which is Sasakian is K-semistable.Proof. In [42] the following inequality of Donaldson [12] is proved for a Sasakianmanifold (
M, η, ξ,
Φ) with polarized cone (
Y, ξ ). For any test configuration of (
Y, ξ )we have(41) inf g ∈S ( ξ, ¯ J ) (Cal( g )) k υ k ξ ≥ c ( n ) Fut( Y , ξ, υ )where c ( n ) > k υ k ξ .We may assume that Φ = ˆ S ( x ) for some x ∈ U . If x is in a polystable orbitthen it admits a csc representative in S ( ξ, ¯ J ) and is thus K-semistable. Thus wemay assume that x is in a non-polystable orbit under the action of G C . If ( Y, ξ )is the polarized cone of ˆ S ( x ) then there is a test configuration Y with π − (1) = Y = ( Y, ξ ) and central fiber Y corresponding to a polystable x ′ ∈ U . Since Y issmooth π : Y → ∆ ⊂ C is a submersion, and by Ehresmann’s theorem there is adiffeomorphism F : C ( M ) × ∆ → Y , where C ( M ) = ( Y, ξ ). Furthermore we may take F to be T -equivariant, where T isthe torus generated by ξ . Then F defines a smooth family of Sasakian structures on M denoted by M z , z ∈ ∆. By Proposition 3.3.2 the central fiber Y = C ( M ) where M is a Sasakian manifold ( M, η , ξ, Φ ) with a cscS deformation in S ( ξ, ¯ J ). Thatis, there is a T -invariant φ ∈ C ∞ b ( M ) so that ˜ η = η + d c φ and ˜ ω T = ω T + dd c φ defines a cscS structure. Using F we see that the deformed structure (˜ η z , ξ, ˜Φ z , ˜ g z )of M z with ˜ η z = η z + d c φ is a Sasakian structure for | z | < ǫ for some small ǫ > η z , ξ, ˜Φ z , ˜ g z ) , | z | < ǫ, z = 0 , is a family of Sasakian structuresin S ( ξ, ¯ J ) with lim z → Cal(˜ g z ) = 0, since the metrics ˜ g z converge uniformly to thecsc structure (˜ η , ξ, ˜Φ , ˜ g o ). The Theorem now follows from (41). (cid:3) Examples
We give some examples of cscS manifolds for which the previous results givenontrivial cscS deformations giving new cscS metrics. We also get some exampleswhich are K-semistable but not K-polystable.
EFORMATIONS OF CSCS METRICS 23
Toric Sasakian manifolds.
We give some toric 5-dimensional cscS manifoldswith nontrivial deformations. The examples are quasi-regular and are given by ex-plicit cones over two dimensional fans describing orbifold toric surfaces. But mod-ifications of the following arguments using nonrational polytopes as in [1] and [28]should give irregular examples also.
Proposition 4.1.1.
Let ( M m +1 , g, η, ξ, Φ) be a compact toric Sasakian manifold.Then H ( M, R ) = 0 and the basic Hodge numbers satisfy h ,kb = 0 for k ≥ .Proof. Let γ ∈ H ,kb be harmonic. Then γ ∈ Γ(Λ k, b ) is a basic, transversely holo-morphic form. The torus T m +1 acts on M preserving the foliation an transverselyholomorphic structure. Let { e i } i =1 ,...,m +1 be a basis of t = Lie( T ) with e m +1 = ξ .At any point of M , in an open dense set, linear combinations of { e i } i =1 ,...,m spanthe transversal holomorphic tangent space. Let f ∧ · · · ∧ f k ∈ Γ(Λ k T M ) where the f i are linear combinations of { e i } i =1 ,...,m . Then γ ( f ∧ · · · ∧ f k ) is constant becauseit basic and transversely holomorphic. But there exists strata where f ∧· · ·∧ f k = 0if k ≥
1, so this constant must be zero.Suppose β ∈ H g is harmonic. Since β is invariant under T , L ξ β = dξ y β = 0.Thus ξ y β = c and an easy argument shows c = 0. Thus H g ∼ = H b = 0, by theabove. (cid:3) If follows from Proposition 3.1.1 that if M is a toric Sasakian manifold H k ( A • ) = H k ( B • ) , for k ≥ . A toric orbifold surface is described by a fan Σ with 1-dimensional conesΣ (1) = { u , . . . , u d } , where u i ∈ Z , i = 1 , . . . , d are not necessarily primitive. Then an orbifold polar-ization is given by a polytope ∆ defined by h u i , x i ≤ λ i , λ i ∈ Z , i = 1 , . . . , d. Denote by X ∆ the polarized surface. Then the vectors w i = ( u i , λ i ) ∈ Z spana cone in Z defining the polarized affine toric variety ( Y, ξ ), with polarization ξ = (0 , ,
1) giving the cone over a toric Sasakian manifold.As in [11] we define a measure dσ on ∂ ∆ which on the edge defined by h u k , x i ≤ λ k is dσ := 1 | u k | dσ , where dσ is the Lebesgue measure. The average of the scalar curvature S of aK¨ahler metric is an invariant of the polarization, and is given by(42) S = R ∂ ∆ dσ R ∆ dµ . S. Donaldson [13] proved that K-polystability with respect to toric degenerationsimplies the existence of a constant scalar curvature K¨ahler metric. In generalK-polystability is difficult to check, but B. Zhou and X. Zhu [48] gave a simplecondition implying K-polystability relative to toric degenerations.Suppose the Futaki invariant of X ∆ vanishes. This is equivalent to the vanishingof L ( θ ) := Z ∂ ∆ θ dσ − S Z ∆ θ dµ for all affine linear functions θ . If(43) S < n + 1 λ i , i = 1 , . . . d, then X ∆ is K-polystable for toric degenerations. Donaldson’s result then impliesthat X ∆ admits a constant scalar curvature K¨ahler metric.Consider CP × CP with the Z q -action α · ([ x , y ] , [ x , y ]) = ([ αx , y ] , [ αx , y ]) , for α ∈ µ q , where µ q ⊂ C ∗ is the group of q-th roots of unity. Then CP × CP / Z q is a toricsurface with fan Σ q given byΣ (1) = { e , e + qe , − e , − e − qe } . For simplicity we restrict to q = 3, shown in Figure 4.1. • • • • •• • • • •• • • • •• • • / / J J ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔ o o (cid:10) (cid:10) ✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔✔ • •• • • • •• • • • •• • • • • Figure 1. CP × CP / Z Then the toric minimal resolution of CP × CP / Z has Σ (1) given by u =(1 , , u = (1 , , u = (1 , , u = (1 , , u = (0 , , u = − u , . . . , u = − u .If one chooses ( λ , . . . , λ ) = (9 , , , , , , , , , S = 1215 + 8 √ √ √ . . . . . The inequalities (43) are clearly satisfied. Then the toric affine variety Y defined bythe w i as above, w = (1 , , , w = (1 , , , . . . , w = (0 , − , M, g, ξ ).Let ˆΣ be the cone spanned by the w i , i = 1 , . . . , d . We are interested in the de-formations of Y ˆΣ preserving ξ = (0 , , H ( B ) = H ( A ) under T H ( A ) = M R ∈W H ( A )( R ) , EFORMATIONS OF CSCS METRICS 25 where clearly each non-trivial term has R vanishing on ξ . The work of N. Iltenand R. Vollmert [24, 46] constructs deformations corresponding to a homogeneouscomponent H ( A )( R ) from admissible Minkowski decompositions of ˆΣ R = ˆΣ ∩{ R =1 } . With R = e ∗ for this example, ˆΣ e ∗ has a three term admissible Minkowskidecomposition ˆΣ e ∗ = ∆ + ∆ + ∆ giving a 2-parameter deformation spanning H ( A )( e ∗ ). Similarly, we have a 2-parameter deformation spanning H ( A )( − e ∗ ). Proposition 4.1.2.
Let M be a toric Sasakian 5-manifold, then H ( B ) = H ( A ) =0 .Proof. Note that H ( A ) = H ∂ b (Λ , ⊗ Λ , ), while the latter consists of trans-versely holomorphic sections and is easily seen to be zero by evaluating a sectionon holomorphic vector fields generated by T . (cid:3) Thus we have a 4-parameter family of deformations H ( A )( e ∗ ) ⊕ H ( A )( − e ∗ )which are integrable by Proposition 4.1.2. The polystable elements with respectto T , the complexification of the identity component of the isometry group of( M, g, ξ ), are (0 ,
0) and ( x , x ) with x = 0 and x = 0. The latter give cscSmetrics with a T group of isometries. The remaining orbits ( x ,
0) and (0 , x ) giveexamples that are K-semistable but not K-polystable by Theorems 3.4.1 and 3.4.2.In the second example we consider a cone over a partial resolution of CP × CP / Z . Let ˆΣ be the cone spanned by w = (1 , , , w = (1 , , , w = (1 , , , w =(0 , , , w = ( − , , , w = ( − , − , , w = ( − , − , , w = (0 , − , Z . Thus Y ˆΣ is a toric affine cone smooth away from the vertex and thecone over a toric Sasakian 5-manifold M .Using (42) we compute S = 20223 + 6 √ √ . . . . and the inequalities (43) are satisfied. And Y ˆΣ is the cone over a toric cscS manifold( M, g, ξ ).Using the same arguments as above and the fact that ˆΣ e ∗ has a two term ad-missible Minkowski decomposition, we have a 2-parameter family of deformations H ( A )( e ∗ ) ⊕ H ( A )( − e ∗ ) . The polystable elements with respect to T are (0 ,
0) and ( x , x ) ∈ H ( A )( e ∗ ) ⊕ H ( A )( − e ∗ ) with x x = 0. The remaining orbits are not K-polystable but areK-semistable.For the third example we consider a non-regular modification of the first example.Define ˆΣ to be the cone spanned by w = (1 , , , w = (1 , , , w = (1 , , , w =(1 , , , w = (0 , , , w = ( − , , , w = ( − , − , , w = ( − , − , , w =( − , − , , w = (0 , − , S = 32265 + 8 √ √ √ . . . . , and inequalities (43) are easily seen to be satisfied, so we have a cscS metric. Inthis example ˆΣ e ∗ has a three term admissible Minkowski decomposition, and weget a 4-parameter family of deformations H ( A )( e ∗ ) ⊕ H ( A )( − e ∗ ) . Again, the polystable orbits are (0 ,
0) and ( x , x ) with x = 0 and x = 0.One can construct an unlimited number of examples by taking cones over partialresolutions of CP × CP / Z q as in these examples.4.2. A 3-Sasakian manifold is a Riemannian manifold(
M, g ) admitting three Sasakian structures ( η i , ξ i , Φ i ) , i = 1 , , ξ i , ξ j ] = − ε ijk ξ k , where ε ijk is antisymmetric and ε = 1. Thus { ξ , ξ , ξ } generate theLie algebra sp (1) of Sp(1). Thus Sp(1) acts by isometries on ( M, g ) rotating theSasakian structures. See [3] and [4] for details.We remark that a 3-Sasakian structure on (
M, g ) is equivalent to a hyperk¨ahlerstructure on ( C ( M ) , g = dr + r g ). Thus M must be of dimension n = 4 m − M, g ) is Sasaki-Einstein with Einstein constant 4 m − η, ξ, Φ) := ( η , ξ , Φ ). Since ( M, g ) has posi-tive Ricci curvature, standard vanishing theorems and Proposition 3.1.1 imply thefollowing.
Proposition 4.2.1.
For ( M, η, ξ, Φ) we have H ( B ) = H ( A ) and H ( A ) = 0 . The element τ = e π j ∈ Sp(1) acts on ( η, ξ,
Φ) by τ · ( η, ξ, Φ) = ( − η, − ξ, − Φ). Weconsider deformations equivariant with respect to τ . We have a conjugate linearisomorphism τ ∗ : H ( A ) → H ( A ) . Since τ = Id, this is a real structure and we define Re H ( A ) to be the subspacefixed by τ ∗ .Let G = Aut( η, ξ, Φ) be the identity component of the automorphism group. Bythe Hilbert-Mumford criterion one gets that the decomposable element of Re H ( A ) ⊗ C are polystable for the action of G C . If one takes a 1-parameter subgroup C ∗ < G C ,then we may assume U(1) ⊂ C ∗ is contained in G by acting by conjugation. Con-sider weight space decomposition H ( A ) = M k ∈ Z V k , and τ ∗ ( V k ) = V − k . Therefore if a decomposable element of Re H ( A ) ⊗ C has anonzero component in V k it also does in V − k . Theorem 4.2.2.
Suppose ( M, g ) is a 3-Sasakian manifold and ( η, ξ, Φ) is a fixedSasakian structure. Then the decomposable elements of Re H ( A ) ⊗ C give integrableSasaki-Einstein deformations.In particular, if H ( A ) = 0 , then ( M, g ) admits a non-trivial Sasaki-Einsteindeformation. In finite families of toric b ( M ) = k ≥ η, ξ, Φ) if b ( M ) > M, η, ξ, Φ) = T (cf. [43]). And one has the following. EFORMATIONS OF CSCS METRICS 27
Proposition 4.2.3 ([43]) . Let ( M, g ) be a toric 3-Sasakian 7-manifold. Then withrespect to a fixed Sasakian structure one has dim H ( A ) = b ( M ) − , and H ( A ) = H ( A ) T . Since T = T C acts trivial on H ( A ), all the elements are trivially polystable.Therefore we get a second proof of a result proved analytically in [45]. Theorem 4.2.4.
Let ( M, g ) be a toric 3-Sasakian 7-manifold. Then ( M, g ) admitsa complex b ( M ) − dimensional space of Sasaki-Einstein deformations. References
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