Deformation of extremal metrics, complex manifolds and the relative Futaki invariant
aa r X i v : . [ m a t h . DG ] M a r DEFORMATION OF EXTREMAL METRICS, COMPLEXMANIFOLDS AND THE RELATIVE FUTAKI INVARIANT
YANN ROLLIN, SANTIAGO R. SIMANCA, AND CARL TIPLER
Abstract.
Let ( X , Ω) be a closed polarized complex manifold, g be an ex-tremal metric on X that represents the K¨ahler class Ω, and G be a compactconnected subgroup of the isometry group Isom( X , g ). Assume that the Fu-taki invariant relative to G is nondegenerate at g . Consider a smooth family( M → B ) of polarized complex deformations of ( X , Ω) ≃ ( M , Θ ) providedwith a holomorphic action of G which is trivial on B . Then for every t ∈ B suf-ficiently small, there exists an h , ( X )-dimensional family of extremal K¨ahlermetrics on M t whose K¨ahler classes are arbitrarily close to Θ t . We apply thisdeformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit K¨ahler-Einstein metrics. Introduction
Every closed K¨ahler manifold is stable under complex deformations. Indeed, theclassical result of Kodaira and Spencer [12] allows us to follow differentiably theK¨ahler metric under small perturbations of the complex structure. Our goal hereis the study of the stability of the extremal condition of a K¨ahler metric undercomplex deformations.Let us recall that a cohomology class Ω ∈ H ( X , R ) on a complex manifold ( X , J )is called a polarization of X if Ω can be be represented by the K¨ahler form ω g ofa K¨ahler metric g on X . In this case, the pair ( X , Ω) is called a polarized complexmanifold. In what follows, we shall sometimes use the metric g and its K¨ahler form ω g interchangeably. We shall denote by c = c ( X , J ) the first Chern class of ( X , J ).The set of all K¨ahler forms on X representing a given polarization Ω is denotedby M Ω . The search for a canonical representative g of Ω is done by looking for acritical point of the functional(1) M Ω → R g Z X s g dµ g . Here, s g is the scalar curvature of g and dµ g its volume form. These critical pointsare the extremal metrics of Calabi [4]. The condition for g to be such can be statedsimply by saying that the gradient of s g is a real holomorphic vector field, which initself shows a subtle interplay between extremal metrics and the complex geometryof X .We denote by Aut( X ) the automorphism group of X . The space h ( X ) of holo-morphic vector fields on X is its Lie algebra. The space h ( X ) of holomorphic vectorfields with zeroes is an ideal of h ( X ). The identity component G ′ = Isom ( X , g ) ofthe isometry group of the metric metric g is identified with a compact subgroup of Aut( X ), and its Lie algebra is denoted g ′ . If g is extremal, then G ′ is a maximalconnected compact subgroup of Aut( X ) [5].1.1. Main result.
Let us consider a polarized complex manifold ( X , Ω) and asmooth family of complex deformations M → B of X ≃ M . Here B is someopen neighborhood of the origin in an Euclidean space R m for some m ≥ § E → B be the vector bundle of second fiber cohomologyof the fibration M → B , whose fibers are E t = H ( M t , R ). A smooth section Θof E → B such that Θ t admits a K¨ahler representative ω t in M t for all t in B issaid to be a polarization of the family M → B . We obtain a family ( M t , Θ t ) of po-larized manifolds parametrized by t ∈ B . A polarized family of complex manifolds( M → B, Θ), and a polarized complex manifold ( X , Ω) together with an isomor-phism X ≃ M that makes Ω and Θ correspond to each other, is said to be apolarized deformation of ( X , Ω).By shrinking the neighborhood of the origin B if necessary, the Kodaira-Spencertheory allows us to choose a smooth 2-form β in M such that ω t = β | M t is a K¨ahlerform in M t representing Θ t . In this expression, β | M t denotes the pullback of β bythe canonical inclusion M t ֒ → M . Such a β is said to represent the polarizationΘ. If g is a K¨ahler metric on X whose K¨ahler form ω g represents Ω, β can beconstructed so that ω and ω g agree under the isomorphism X ≃ M .Let us now assume that the metric g is extremal. It is then a natural to ask ifthe representative β of the polarization Θ can be chosen so that the metric g t , ofK¨ahler form ω t on M t , is extremal.A positive answer to this general statement is not to be expected, and as it hasbeen pointed out in [14], “the answer is an emphatic no ” (cf. [3] for some counter-examples). However, if we assume some symmetries for M and the nondegeneracyof the relative Futaki invariant, the answer is actually yes provided we allow thepolarization Θ to be deformed also.If the metric g on X is extremal, then G ′ = Isom ( X , g ) is a maximal connectedcompact subgroup of Aut( X , g ) that acts holomorphically on the central fiber M of M → B . In general though, this action will not extend as a holomorphic action of G ′ (cf. § M → B . We shall assume that the action of G ′ extends partially,and we have a connected compact subgroup G of G ′ that acts holomorphically on M → B with trivial action on B . We denote by C ∞ G ( M ) the space of G -invariantsmooth functions on M .It is then possible to introduce the notion of reduced scalar curvature s Gg for any G -invariant K¨ahler metric on X (cf. § G (cf. § F cG, Ω : q / g → R , where g is the Lie algebra of G and q is the normalizer of g in h ( X ). The extremalcondition of a metric is encoded in the equation s Gg = 0, and a G -invariant extremalmetric representing Ω have vanishing reduced scalar curvature if, and only if, F cG, Ω vanishes identically [8, 5]. This characterization of extremal metrics reinterprets inthis manner a presentation advocated elsewhere [16, 17, 20]. EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 3
Let p be the normalizer of g in g ′ . We set g = g ∩ h ( X ) and p = p ∩ h ( X ),respectively. The differential of the relative Futaki invariant induces a linear map q / g → ( H , ( X ) ∩ H ( X , R )) ∗ . We say that F cG, Ω is nondegenerate if the restriction of this map to p / g is injective.Using the holomorphic action of G on M → B , we refine the construction of therepresentative β of the polarizatin Θ so that the K¨ahler metric g t it induces on M t is G -invariant (cf. § t ∈ B . We then use the corresponding K¨ahlerform ω t as the origin of the affine space of K¨ahler metrics on M t that representthe polarization Θ t . Let H t be the space of real g t -harmonic (1 , M t .By the Kodaira-Spencer theory, and perhaps shrinking B to a sufficiently smallneighborhood of the origin, the spaces H t are the fibers of a smooth vector bunde H t → B . Given a section α of the bundle H → B and a sufficiently small function φ ∈ C ∞ G ( M ), we consider the family of K¨ahler metrics g t,α,φ on M t defined by theK¨ahler forms ω t,α,φ = ω t + α | M t + dd c φ | M t . Assuming that g = g is extremal, we seek solutions g t,α,φ of the equation s Gg t,α,φ = 0for values of ( t, α, φ ) in a sufficiently small neighborhood of (0 , ,
0) where thesolution takes on the value g , , = g . Introducing suitable Banach spaces, the saidequation defines as Fredholm map that is a submersion at the origin if, and only if,the relative Futaki invariant is nondegenerate. The implicit function theorem thenyields our main result: Theorem A.
Let ( M → B, Θ) be a polarized family of deformations of a closedpolarized complex manifold ( X , Ω) . Suppose that g is an extremal metric whoseK¨ahler form ω g respresents Ω , and that G is a compact connected subgroup of G ′ = Isom ( X , g ) such that • G acts holomorphically on M → B and trivially on B , • the reduced scalar curvature s Gg of g is zero, • the Futaki invariant relative to G is nondegenerate at g .Then, given any G -invariant representative β of the polarization Θ such that inducedmetric g on M agrees with g via the isomorphism M ≃ X , and shrinking B to asufficiently small neighborhood of the origin if necessary, the space of K¨ahler metricson M t with vanishing reduced scalar curvature lying sufficiently close to the metric g t induced by β is a smooth manifold of dimension h , ( X ) . In particular, there arearbitrarily small perturbations Θ ′ of the polarization Θ such that Θ ′ t is representedby an extremal metric. Remark 1.1.1 . Theorem 4.4.3 is a more precise and technical version of Theo-rem A. In particular, it contains more informations about the kind of deformationsΘ ′ of the polarization Θ. This refined version of the above theorem will lead to theapplications below.1.2. Applications.
Theorem A generalizes the results of [13, 14]. The case G = { } corresponds to the deformation theory of constant scalar curvature K¨ahler met-rics whereas the case G = G ′ = Isom ( X , g ) corresponds to the deformation theoryof extremal metrics (cf. § Y. ROLLIN, S.R. SIMANCA, AND C. TIPLER choices for the group G is new. We illustrate the power of Theorem A with theanalysis of some new examples of extremal metrics, and some cases of interest.However, we observe that the applicability of our theorem to these follows by ele-mentary reasons, and the nondegeneracy of the relative Futaki invariant is easy tosee. It would be of interest to find applications of our result in more demanding sit-uations, which could illustrate the problem at hand in further detail. Some highlynontrivial applications were already given in [14].1.2.1. Maximal torus symmetry.
Extremal K¨ahler metrics are automatically stableunder complex deformations with maximal torus symmetry:
Corollary B.
Let ( M → B, Θ) be a polarized family of deformations of a closedpolarized complex manifold ( X , Ω) . Assume that Ω admits an extremal representa-tive and that M → B is endowed with a holomorphic action of a maximal compacttorus G = T ⊂ Aut( X ) acting trivially on B . Then for t ∈ B sufficiently small, Θ t is represented by an extremal K¨ahler metrics on M t . Corollary B may look reminiscent of [1, Lemma 4], where a stability result ofthe extremal condition under complex deformations with symmetries is obtained asan extension of the theory of [13]. This result is carried out under such restrictiveassumptions on the deformation of the complex structure and the K¨ahler class thatthe scope of its applicability is rather limited.1.2.2.
The Mukai-Umemura -fold. We may apply Theorem A to the study of theMukai-Umemura Fano 3-fold X , with automorphism group Aut( X ) = PSL (2 , C ).Donaldson has proven that this variety admits a K¨ahler-Einstein metric [7]. Ourdeformation theorem applies. We obtain the following result: Corollary C.
Let ( M → B, Θ) be a polarized deformation of the Mukai-Umemura -fold with polarization ( X , c ( X )) and Θ t = c ( M t ) . Assume M → B is one ofthe deformations described at § , endowed with a holomorphic action of a group G ⊂ PSL (2 , C ) isomorphic to a dihedral group of order or a semidirect product S ⋊ Z / . Then for every t ∈ B sufficiently small, Θ t is represented by a K¨ahler-Einstein metric on M t . The result above was proved already in [7] using a different approach whichwas later refined by Sz´ekelyhidi [21]. Using this method it is possible to understandgeneral small complex deformations of the Mukai-Umemura 3-fold. It turns out thatsmall deformations corresponding to polystable orbits under the action of
PSL (2 , C )are exactly the one which carry Kaehler-Einstein metrics.1.3. Plan of the paper. In § § §
4, where we prove an expanded version of our main Theorem A. The applicationsare given in §
5. We begin in § § § EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 5
Acknowledgments.
We would like to thank Professor Simon Donaldson forpointing out an error in the attempt to apply our deformation theory to the Mukai-Umemura 3-fold in an earlier version of this work. We also thank Professor PaulGauduchon for making the preliminary version of his book [10] available to us.2.
Extremal metrics
Let ( X , Ω) be a closed polarized complex manifold of dimension n . The polarizingassumption on the class Ω may be stated by saying that there exist K¨ahler metrics g whose K¨ahler form ω g represents it. We let M Ω be the set of all such metrics. If g ∈ M Ω , the projection of the scalar curvature s g onto the constants is given by(2) s Ω = 4 nπ c ∪ Ω n − Ω n , and there exists a vector field X Ω ∈ h ⊂ h [8, 9] such that for every g ∈ M Ω wehave(3) Z X s g dµ g ≥ s Ω n n ! − F ( X Ω , Ω) . The equality is achieved if, and only if, the metric is extremal. This lower boundwas known earlier [11, 18] for metrics in M Ω that are invariant under a maximalcompact subgroup of Aut( X ). It was proven by Chen [6] to hold in general, for anymetric in M Ω . The lower bound varies smoothly as a function of the polarizingclass Ω [18].2.1. Holomorphic vector fields.
The subset h of holomorphic vector fields withzeroes is an ideal of h , and the quotient algebra h / h is Abelian. A smooth complex-valued function f gives rise to the (1 , f ∂ g f defined by theexpression g ( ∂ g f, · ) = ∂f . This vector field is holomorphic if, and only if ∂∂ g f = 0, a condition equivalent to f being in the kernel of the Lichnerowicz operator (4) L g f := ( ∂∂ g ) ∗ ∂∂ g f = 14 ∆ g f + 12 r µν ∇ µ ∇ ν f + 12 ( ∇ ℓ s g ) ∇ ℓ f . The ideal h consists of vector fields of the form ∂ g f , for a function f in thekernel of L g . Or put differently, a holomorphic vector field Ξ can be written as ∂ g f if, and only if, the zero set of Ξ is nonempty [14]. The kernel H g of L g iscalled the space of holomorphy potentials of ( X , g ). Since L g is elliptic, H g is finitedimensional complex vector space and consists of smooth functions.Generally speaking, the Lichnerowicz operator L g of a metric g is not a realoperator, and so the real and imaginary parts of a function in its kernel do not haveto be elements of the kernel also. In studying the geometry of K¨ahler manifolds, it isoften convenient to use real quantities and operators. We introduce the terminologythat allows us to do so here. We follow the conventions of [2, 10].Let us consider the operator d c on functions defined by d c = Jd . Given a realholomorphic vector field Ξ on X , the Hodge theory provides a unique decomposition ξ = Ξ ♭ = ξ h + du Ξ + d c v Ξ , where ξ h is a harmonic 1-form, u Ξ , v Ξ are real valuedfunctions, and d c v Ξ is coclosed. The functions u Ξ , v Ξ are uniquely determined upto an additive constant. The dual of this identity gives rise to the decompositionΞ = Ξ h + grad u Ξ + J grad v Ξ , and it follows that Ξ , = (Ξ − iJ Ξ) = Ξ , h + ∂ g f Ξ , Y. ROLLIN, S.R. SIMANCA, AND C. TIPLER where f Ξ = u Ξ + iv Ξ . It follows that Ξ is real holomorphic if, and only if, Ξ , iscomplex holomorphic, that is to say, if, and only if, L g f Ξ = 0. If we extend theLie bracket operation linearly in each component, the integrability of the complexstructure makes of the map Ξ → Ξ , an isomorphism of Lie algebras.As indicated above, the condition Ξ h = 0 is equivalent to Ξ being Hamiltonian,and characterizes h ( X ). If Ξ ∈ h ( X ), the potential f Ξ of Ξ is a solution of theequation L Ξ , ω g = dd c f . On the other hand, a holomorphic vector field Ξ isparallel if, and only if, Ξ is the dual of a harmonic 1-form.A vector field Ξ is a Killing field if, and only if, it is holomorphic and its potentialfunction f Ξ = iv Ξ is, up to a constant, a purely imaginary function [14]. In thatcase, v Ξ will be called the Killing potential of Ξ.If ξ is a 1-form, we let ∇ − ξ be the J anti-invariant component of the 2-tensor ∇ g ξ . Let ξ = Ξ ♭ for a real vector field Ξ. The condition ∇ − ξ = 0 is equivalent toΞ being a real holomorphic vector field, and can be expressed as δδ ∇ − ξ = 0.We have the identities(5) δ ∇ − ξ = 12 ∆ g ξ − Jξ ♯ ρ g and(6) δδ ∇ − ξ = 12 ∆ g δξ − h d c ξ, ρ g i + 12 h ξ, ds g i , respectively.We introduce the operator L g defined by L g f = ( ∇ − d ) ∗ ( ∇ − d ) f = δδ ∇ − df . This is a real elliptic operator of order four, and if f is a real valued function, wehave that L g f = 0 if, and only if, grad f is a real holomorphic vector field; bythe observations above, every Hamiltonian Killing field is of the form J grad f for afunction f of this type.By (6) we derive the identities(7) L g f = ∆ g f + h dd c f, ρ g i + h df, ds g i ,δδ ∇ − d c f = − L J grad s g f , and by (4), we see that(8) 2 L g f = L g f + i L J grad s g f . Lemma 2.1.1.
For any K¨ahler metric g , the space of real solutions of the equation L g f = 0 coincide with the space of real solutions of L g f = 0 .Proof. By (8), if f is a real valued function such that L g f = 0, then L g f = 0and L J grad s g f = 0. Conversely, let us assume that L g f = 0. This implies thatΞ = J grad f is a Killing field. Then L J grad s g f = h d c s g , df i = −h ds g , d c f i = − L Ξ s g = 0 , and by (8), we conclude that L g f = 0. (cid:3) The group of isometries of a K¨ahler metric.
Let ( X , Ω) be a polarizedcomplex manifold and G , G ′ be connected compact Lie subgroups of Aut( X ), with G ⊂ G ′ . By taking a G ′ -average if necessary, we represent the polarization Ω theK¨ahler form ω g of a G ′ -invariant K¨ahler metric g . We attach to this data a relativeFutaki invariant. EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 7
Lie algebras.
The Lie algebras g and g ′ of G and G ′ are naturally identifiedwith subalgebras of the algebra h ( X ) of holomorphic vector fields. We define g = g ∩ h ( X ) and g ′ = g ′ ∩ h ( X ). We then introduce the Lie algebras • z = Z ( g ), the center of g , • z ′ = C g ′ ( g ), the centralizer of g in g ′ , • z ′′ = C h ( g ), the centralizer of g in h , • p = N g ′ ( g ), the normalizer of g in g ′ , and • q = N h ( g ), the normalizer of g in h .If t is any of these Lie algebras, we shall denote by t the ideal of Hamiltonianvector fields t = t ∩ h ( X ), and by H t g the corresponding spaces of holomorphypotentials.The space of holomorphy potentials H z g (respectively H z ′ g , H z ′′ g ) is identified tothe G -invariant potentials of H g g (respectively H g ′ g , H g ). Notice that H z g ⊂ H g g and H z ′ g ⊂ H g ′ g consists of purely imaginary functions whose imaginary parts defineKilling potentials.Now z is an ideal of z ′ and z ′′ . On the other hand, g is an ideal of p and q . Wehave canonical injections(9) z ′ / z ֒ → p / g , z ′ / z ֒ → p / g , z ′′ / z ֒ → q / g , z ′′ / z ֒ → q / g . Lemma 2.2.2.
The canonical injections (9) are surjective, and we have canonicalisomorphisms of Lie algebras z ′ / z ≃ p / g , z ′ / z ≃ p / g , z ′′ / z ≃ q / g , z ′′ / z ≃ q / g . Proof.
We prove that p = z ′ + g . Let Ξ be a Killing field in g ′ . We have thatΞ = Ξ h + J grad v Ξ where Ξ h is the dual vector field of a harmonic 1-form, and v Ξ is a real function.The vector field Ξ belongs to p if, and only if,(10) [ Y, Ξ] = L Y Ξ ∈ g for all Y ∈ g .Since L Y Ξ h = 0 because Y is Killing, this condition is equivalent to L Y J grad v Ξ ∈ g .In turn, this is equivalent to having J grad ( Y · v Ξ ) ∈ g since Y preserves J and themetric, which means that Y · v Ξ ∈ i H g g for every Y ∈ g . This implies that for every γ ∈ G , we have γ ∗ v Ξ − v Ξ ∈ i H g g . We can then average the function v Ξ under thegroup action to obtain a G -invariant function ˜ v Ξ on X such that ˆ v = v Ξ − ˜ v Ξ ∈ i H g g ,and Ξ = (Ξ h + J grad ˜ v Ξ ) + J grad ˆ v , where J grad ˆ v ∈ g and Ξ h + J grad ˜ v Ξ = Ξ − J grad ˆ v ∈ g ′ is G -invariant and so anelement of z ′ .An analogous argument shows that q = z ′′ + g , which finishes the proof. (cid:3) The reduced scalar curvature.
Let L k ( X ) be the k th Sobolev space defined by g . The space L k,G ( X ) of G -invariants functions in L k ( X ) can be obtained as metriccompletion of C ∞ G ( X ). If k > n , L k ( X ) is a Banach algebra. In fact, the Sobolevembedding theorem says that if k > n + l we have L k ( X ) continuously containedin C l ( X ), the space of functions with continuous derivatives of order at most l . Weshall work below always imposing this restriction over k . Y. ROLLIN, S.R. SIMANCA, AND C. TIPLER
The L -Hermitian product on L k,G ( X ) induced by the Riemannian metric g allows us to define the orthogonal W k,g of i H z g . Thus we have an orthogonaldecomposition L k,G ( X ) = i H z g ⊕ W k,g together with the associated projections π Wg : L k,G ( X ) → W k,g and π Gg : L k,G ( X ) → i H z g . We introduce the reduced scalar curvature s Gg defined by s Gg = π Wg ( s g ) = (1l − π Gg )( s g )By construction, the condition(11) s Gg = 0is equivalent to s g ∈ i H z g , and since H z g ⊂ H g , this condition implies the extremal-ity of the metric g .2.2.4. The reduced Ricci form.
Let L Λ , k,G ( X ) be the space of real G -invariant(1 , X in L k . We lift the projection π Gg to the a projectionΠ Gg : L Λ , k +2 ,G ( X ) → L Λ , k,G ( X )defined by Π Gg β = β + dd c f , where f = G g ( π Wg ( ω g , β )) and G g is the Green operator of g [19, 16, 20]. It followsthat ( ω g , Π Gg ( β )) = π Gg ( ω g , β ).We apply this projection map to the Ricci form, and define the reduced Ricciform [16] ρ G := Π G ρ . We then obtain the identity(12) ρ g = ρ G + 12 dd c ψ Gg , where ψ Gg = 2 G g ( π Wg ( ω g , ρ )) = G g π Wg ( s g ) = G g ( s Gg ). In particular, ρ G = ρ g if, andonly if, s Gg = 0.2.2.5. Variational formulas.
Given a K¨ahler metric g , we consider infinitesimal de-formations of it given by ω t = ω g + tα + tdd c φ of the K¨ahler class ω g , where φ is a smooth function and α is a g -harmonic (1 , ω g ] + t [ α ]. Inorder to avoid confusion, the derivative at g of various geometric quantities will bedenoted by D g below. Sometime later, and when confusion cannot occur, these samederivatives will be denoted by a dot superimposed on the quantities themselves.The holomorphy potential of a holomorphic vector field depends upon the choiceof the metric. Its infinitesimal variation when moving the metric in the directionof φ or α is described in the following lemma: EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 9
Lemma 2.2.6.
Let Ξ be a real holomorphic vector field in h ( X ) , with holomorphypotential f Ξ = u Ξ + iv Ξ ∈ H g . The variation D g f Ξ ( φ ) of f Ξ at g in the directionof φ is given by D g f Ξ ( φ ) = 2 L Ξ , f Ξ , which is equivalent to the expressions ( D g u Ξ )( φ ) = L Ξ φ and ( D g v Ξ )( φ ) = − L J Ξ φ for the variations of the real and imaginary parts of f Ξ . The variations D g u Ξ ( α ) and D g v Ξ ( α ) at g in the direction of a trace-free form α are given by D g u Ξ ( α ) = − G g ( δ ( J Ξ α )) ,D g v Ξ ( α ) = G g ( δ (Ξ α )) . The variation of the reduced scalar curvature s Gg is described by: Lemma 2.2.7.
Let g be a G -invariant K¨ahler metric on X . The variation of thereduced scalar curvature s Gg when moving the metric in the direction of φ is givenby ( D g s Gg )( φ ) = − L g φ + h dφ, ds Gg i . If s Gg = 0 , the variation of s Gg when varying the metric in the direction of a trace-freeform α is given by ( D g s Gg )( α ) = π W ( G g ( h α, dd c s g i ) − h α, ρ g i ) . The relative Futaki invariant.
Given any G -invariant K¨ahler metric g on X with K¨ahler form ω g and Ricci-form ρ g , we define the Futaki function of ( G, g )by F cG,ω g (Ξ) = Z X d c ψ Gg (Ξ) dµ g = Z X − L J Ξ ψ Gg dµ g for any real holomorphic vector field Ξ on X . Here, ψ Gg is the Ricci potential (12)of the metric g . This function is defined in terms the K¨ahler metric g , but dependsonly upon the K¨ahler class Ω = [ ω g ] [8, 5], as we briefly recall below. The resultingfunction F cG, Ω shall be referred to as the Futaki invariant of Ω relative to G , orrelative Futaki invariant for short, when the group G and class Ω are understood.Let us observe that the usual definition of the real Futaki invariant applied to aholomorphic vector field Ξ is given by F G, Ω (Ξ) = F cG, Ω ( J Ξ). We have introducedthis notation for convenience. Put differently, there is a complex valued version ofthe Futaki invariant such that F G, Ω is its real part and F cG, Ω the imaginary part.2.3.1. Properties of the Futaki invariant.
The function F cG,ω g (Ξ) vanishes on par-allel holomorphic vector fields. Indeed, let Ξ h be the dual of a harmonic 1-form ξ h .Then F cG,ω (Ξ h ) = Z X h Jξ h , dψ Gg i dµ g = Z X h δJξ h , ψ Gg i dµ g . Since the space of harmonic 1-forms is J -invariant, so δJξ h = 0, and it follows that F cG,ω g (Ξ h ) = 0.This function F cG,ω g can be expressed alternatively in terms of the reduced scalarcurvature. For if Ξ is a Hamiltonian holomorphic vector field on ( X , g ), it can bewritten as Ξ = grad u Ξ + J grad v Ξ for some real valued functions u Ξ , v Ξ , and with d c u Ξ orthogonal to the space of closed forms. Hence, F cG,ω (Ξ) = h dv Ξ , dψ Gg i = h v Ξ , δdψ Gg i = h v Ξ , ∆ g ψ Gg i . It follows that(13) F cG,ω g (Ξ) = F cG,ω g (grad u Ξ + J grad v Ξ ) = Z X v Ξ s Gg dµ g . The expression above shows that if Ξ is any Killing field in g , then F cG,ω (Ξ) = 0.It follows that F cG,ω g induces a R -linear map(14) F cG,ω g : q / g → R , where q is the normalizer of g in h ( X ).The fundamental properties of this function is now stated in the form of a Theo-rem. This was proven originally by Futaki [8], and generalized a bit later by Calabi[5]. The version given here follows its presentation in [10]. Theorem 2.3.2.
The relative Futaki function F cG,ω g : q / g → R defined in (14) isindependent of the particular G -invariant K¨ahler representative ω g of the class Ω ,and so it induces an invariant function F cG, Ω : q / g → R of the class, the relative Futaki invariant of Ω . The relative Futaki invariant van-ishes for a class Ω if, and only if, any G -invariant extremal metric g that represents Ω has vanishing reduced scalar curvature s Gg . If g is any G -invariant extremal K¨ah-ler metric and G ′ = Isom ( X , g ) , the vanishing of F cG, Ω is equivalent to the vanishingof its restriction F cG, Ω : p / g → R . Proof . The invariance is proven in [8], and generalized in [5, Proposition 4.1, p.110].The proof of the second statement follows by (13). For if s Gg = 0, then F cG,ω (Ξ) =0 for every Ξ ∈ q . Conversely, let us assume that F cG, Ω vanishes on q and, therefore,on q = q ∩ h ( X ). Let g be a G -invariant extremal K¨ahler metric that representsthe class Ω, and let G ′ = Isom ( X , g ). Then Ξ = J grad s g ∈ g ′ , and by [5, Theorem1, p. 97], L Ξ Z = 0 for all Killing fields Z . Thus, we have that Ξ ∈ p ⊂ p ⊂ q . But F cG, Ω (Ξ) = 0 = Z X s g s Gg dµ g = k s Gg k g , and so s Gg = 0.The final statement follows easily by the argument above. (cid:3) Nondegeneracy of the Futaki invariant.
Let g be a metric such that [ ω g ] = Ω.Given any real g -harmonic (1 , g to compute thederivative ddt F cG, Ω+ tα | t =0 . We have the following:
Lemma 2.3.4.
Let g be a G -invariant extremal K¨ahler metric on X such that [ ω g ] =Ω , and let α be a real g -harmonic trace-free (1 , -form. If Ξ = grad u Ξ + J grad v ΞEFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 11 is a holomorphic vector field in q , then ddt F cG, Ω+ tα | t =0 (Ξ) = Z X v Ξ ( D g s Gg )( α ) dµ g = Z X π W ( v Ξ ) ( G g ( h α, dd c s g i ) − h α, ρ g i ) dµ g . Proof . Since α is trace-free, the infinitesimal variation of the volume form iszero. We use this fact in the differentiation of the relative Futaki invariant givenby expression (13), and obtain that ddt F cG, Ω+ tα | t =0 (Ξ) = Z X ˙ v Ξ ( α ) s Gg dµ g + Z X v Ξ ˙ s Gg ( α ) dµ g = Z X v Ξ ˙ s Gg ( α ) dµ g where the last equality follows because the reduced scalar curvature s Gg of g vanishes.We then use Lemma 2.2.7 to obtain that ddt F cG, Ω+ tα | t =0 (Ξ) = Z X π W ( v Ξ )( G g ( h α, dd c s g i ) − h α, ρ g i ) , which finishes the proof. (cid:3) Notice that we can state the result above using ρ Gg instead because ρ g = ρ Gg formetric g with vanishing reduced scalar curvature.The differential of the relative Futaki invariant of Lemma 2.3.4 defines a linearmapping(15) q / g → ( H , ( X ) ∩ H ( X , R )) ∗ . This leads to a very important concept in our work:
Definition 2.3.5 . Let G and G ′ be connected compact subgroups of Aut( X ) suchthat G ⊂ G ′ . The Futaki invariant F cG, Ω relative to G is said to be G ′ -nondegenerateat Ω if the linear map (15) restricted to p / g ≃ z ′ / z → ( H , ( X ) ∩ H ( X , R )) ∗ is injective. If g is a K¨ahler metric representing Ω such that G ′ = Isom ( X , g ) andthis condition holds for some G ⊂ G ′ , we say that g is Futaki nondegenerate relativeto G .We briefly illustrate this notion in a particular case next.2.3.6. Nondegeneracy of the Futaki invariant relative to a maximal compact torus.
In the presence of certain maximal torus symmetries on the underlying manifold,the relative Futaki invariant vanishes.
Lemma 2.3.7.
Let X be a complex manifold, and let T be a maximal compacttorus subgroup of Aut( X ) . Then for G = T , we have q / g = 0 and the relativeFutaki invariant F cG, Ω is identically zero for any K¨ahler class Ω in X . Given anycompact connected Lie group G ′ ⊂ Aut( X ) , then for G = T ⊂ G ′ the relative Futakiinvariant F cG, Ω is G ′ -nondegenerate.Proof. The Lie algebra g of G = T is Abelian and so equal to its center z . Thecentralizer z ′′ of g in h contains z . If z were strictly contained in z ′′ , we could findan element of z ′′ \ z that together with z would generate an Abelian Lie algebra,and this would contradict the maximality of z in h . Thus, z = z ′′ , and by Lemma2.2.2, q / g ≃ z ′′ / z = 0. The result follows. (cid:3) Deformations of the complex structure
In this section we recall the theory of smooth deformations of a complex manifoldand that of smooth polarized deformations. We illustrate the concept exhibiting adeformation of the Hirzebruch surface F . This deformation will reappear in one ofour applications in § Complex deformations.
A smooth family of complex deformations consistsof the following data:(1) an open connected neighborhood B of the origin in R m , a smooth manifold M and a smooth proper submersion ϕ : M → B ,(2) an open covering { U j } j ∈ I of M and smooth complex valued functions z j =( z j , · · · , z nj ) defined on each U j , such that the collection of mappings U j ∩ ϕ − ( t ) → C n p ( z j ( p ) , . . . , z nj ( p ))define a holomorphic atlas on each manifold ϕ − ( t ).Such a family of complex deformations will be denoted simply by M → B , and thefibers ϕ − ( t ) together with their canonical complex structure by M t . A complexmanifold X and a family of deformations M → B together with a given isomorphism X ≃ M is called a complex deformation of X .Smooth families of complex deformations are smoothly locally trivial. Indeed,let M be the underlying differentiable manifold of the central fiber M of a family M → B . At the expense of shrinking B if necessary, we can find a diffeomorphism M → B × M such that the diagram M ❅❅❅❅❅❅❅❅ / / B × M π B { { ✇✇✇✇✇✇✇✇✇ B commutes. Here π B denotes the projection map onto the first factor. We refer tothis diffeomorphism as a trivialization of the given family of complex deformations.By way of such a trivialization, we see that all the M t s are diffeomorphic to M for t s that are in a sufficiently small neighborhood of the origin in R m , and thatthe family of complex manifolds M t can be seen as a differentiable family { J t } of integrable almost complex structures on M . From this point of view, M t and( M, J t ) are identified as complex manifolds.3.2. Polarized deformations.
Let us now assume that ( X , Ω) is a polarized com-plex manifold, and that ( M → B, Θ) is a polarized complex deformation of it. If ω g is a K¨ahler form on X that represents Ω, a deep result of Kodaira and Spencer[12] shows that we can find a smooth family of K¨ahler metrics on M t that extends ω g and represents Θ t for each t (cf. § { J t } on M that a trivialization of the deformation allows, thefunction t → h p,qt = dim C H q ( M, Ω pJ t ) is upper semi-continuous and if ( M, J, ω g ) isK¨ahler, this function is in fact constant in a sufficiently small neighborhood of theorigin. Then it follows that there exists a smooth family t → ω t of 2-forms on M such that ω t is K¨ahler with respect to J t , [ ω t ] = Θ t ∈ H ( M ; R ), and ω and ω g agree with each other via the identification X ≃ M ≃ ( M, J ). This point of view EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 13 is best adapted to our work here. We obtain a 2-form β on M such that β | M t = ω t .Such a form β is said to represent the polarization Θ.3.3. Example: the Mukai-Umemura -fold. Let V be a 7-dimensional complexvector space, Gr ( V ) be the Grassmanian of complex 3-dimensional subspaces of V and U → Gr ( V ) be the tautological rank 3-bundle over Gr ( V ). Notice that Gr ( V ) is 12-dimensional.Any ̟ ∈ Λ V ∗ defines an section σ ̟ of the bundle Λ U ∗ → Gr ( V ). Let Z ̟ ⊂ Gr ( V ) be the zero set of σ ̟ . So Z ̟ is the subset of isotropic 3-planes of ̟ , the points P in Gr ( V ) such that ̟ | P = 0. For a generic ̟ , Z ̟ is a smoothsubvariety of codimension three, and given three linearly independent forms, ̟ , ̟ , ̟ , we obtain the 3-fold X Π = X ̟ ,̟ ,̟ = Z ̟ ∩ Z ̟ ∩ Z ̟ ⊂ Gr ( V ) , that depends only on the 3-plane Π spanned by the ̟ i s in Λ V ∗ , and not in thebasis chosen to represent it. The action of the group SL ( V ) on V induces an actionon Gr (Λ V ∗ ), and 3-planes Π and Π define isomorphic complex varieties X Π and X Π if, and only if, the planes Π and Π lie in the same SL ( V ) orbit. Weobtain a set of equivalence classes of 3-folds parametrized by the quotient U / SL ( V ).There is a Zariski open set C ⊂ Gr (Λ V ∗ ) of 3-planes Π such that X Π is asmooth subvariety of dimension 3. It has an obvious family of complex deforma-tions. For if N = { (Π , x ) ∈ C × Gr ( V ) | x ∈ X Π } , we may consider this smoothcomplex manifold together with its canonical projection N → C . The SL ( V )-actionon C × Gr ( V ) induces an equivariant action on N → C , and two fibers are iso-morphic if, and only if, they are above the same orbit in C .The Mukai-Umemura manifold is a particular smooth 3-fold in this family. It canbe described efficiently as follows. The six symmetric power Sym ( C ) is the stan-dard irreducible 7-dimensional representation of SL (2 , C ). We take V = Sym ( C )with its induced SL (2 , C )-action. The representation Λ V ∗ decomposes into irre-ducible representations asΛ (Sym ( C )) = Sym ( C ) ⊕ Sym ( C ) ⊕ Sym ( C ) , The summand Sym ( C ) corresponds to a 3-plane Π in Λ V ∗ that defines theMukai-Umemura variety X Π . The plane Π is invariant under SL (2 , C ), so thisgroup acts naturally on X Π .Since many of the deformations are equivalent via the SL ( V )-action, it is im-portant to describe the quotient of Gr (Λ V ∗ ). This is done carefully in [7], whereit is proven that the quotient of the tangent space to Gr (Λ V ∗ ) at Π by thetangent space to the the orbit can be identified with Sym ( C ) with its standard SL (2 , C )-action, the stabilizer of Π in SL ( V ). By the theory for equivariant slicesof Lie group actions, there is an neighborhood of the origin B ⊂ Sym ( C ) and a PSL (2 , C )-equivariant embedding j : B → Gr (Λ V ∗ ) such that for t , t ∈ B , theimages j ( t ) and j ( t ) are in the same SL ( V )-orbit if, and only if, t and t are inthe same PSL (2 , C ) orbit. If B is taken to be sufficiently small, we have j ( B ) ⊂ C and M → B , defined as the fiber product M = B × C N , is a smooth family ofcomplex deformations of M ≃ X Π .3.3.1. Deformations with symmetries.
In particular, the deformations correspond-ing to polynomials p = C ( u − αv )( v − αu ) for α, C ∈ C × have a stabilizer G in PSL (2 , C ) isomorphic to a dihedral group of order 8. In the case where α = 0, the stabilizer of p is the subgroup of PSL (2 , C ) spanned by the one parameter subgroup λ · [ u : v ] = [ λu : λ − v ] and the rotation [ u : v ] [ u : v ]. Hence, we have a maximalcompact subgroup G of the stabilizer of p that is isomorphic to Z / ⋊ S .We can consider the family of deformation of X Π obtained by restricting B tothe subspace of polynomials of the form tp as above for t ∈ C . Thus, we get afamily of deformation M → C endowed with a holomorphic action of G , where G is a dihedral group of order 8, or the semidirect product of Z / ⋊ S in the casewhere α = 0. In addition, the group G acts on the central fiber as a subgroup of PSL (2 , C ), the identity component of the automorphism group of X Π . Therefore, G acts trivially on the cohomology of every fiber M t of the deformation M → C .4. Deformations of extremal metrics
In this section we prove a criterion that ensures the stability of the extremalcondition of a K¨ahler metric under complex deformations. We assume some sym-metries of the family of deformations and the nondegeneracy of the relative Futakiinvariant.Let M → B be a smooth family of complex deformations of a complex manifold X ≃ M . If we assume that X is of K¨ahler type, then it follows that all fibers M t are K¨ahler provided we shrink the set of parameters t ∈ B to a sufficiently smallneighborhood of the origin [12]. In particular, the Lie algebra of holomorphic vectorfields h ( M t ) contains the ideal h ( M t ) consisting of holomorphic vector fields withzeroes somewhere (cf. § X is of K¨ahler type and that B has been so restricted. We shall indicate the occasionswhere the latter restriction may be necessary.4.1. Holomorphic group actions.
Let G be a compact connected Lie groupacting smoothly on M such that: • The fibers M t are preserved under the action. • The induced action on each M t is holomorphic, • G acts faithfully on M , and it is identified to a subgroup of the connectedcomponent of the identity of Aut( M ).Under this conditions, we say that G acts holomorphically on M and trivially on B . As discussed in §
3, we may think of the deformation M → B as a smoothlyvarying family of integrable almost complex structure J t on the underlying manifold M of the central fiber such that ( M, J t ) ≃ M t , and using the smooth trivialisationof the deformation near the central fiber, the holomorphic G -action on M can beseen as a smoothly varying family of G -actions a : B × G × M → M ( t, g, x ) a t ( g, x )where a t is a J t -holomorphic action that is identified to the G -action on M t modulothe isomorphism ( M, J t ) ≃ M t .By [15], we know that compact Lie group actions are rigid up to conjugation.Hence, there exists a smooth isotopy f t : M → M that intertwines the a t and a actions, f − t ◦ a t ( g, f t ( x )) = a ( g, x ) for all t ∈ B , g ∈ G , x ∈ M , EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 15 possibly after restricting B to some smaller neighborhood of the origin. So actingby f t on the family of complex structures J t , we may assume that the action of G is independent of t , and that it is holomorphic relative to each J t .The triviality of the smooth deformation of the action of G on M has some strongconsequences on the complex geometry. Firstly, there is a canonical morphism ofthe Lie algebra g of the group G into the space of smooth vector fields on M , ξ : g ֒ → C ∞ ( M, T M ) . a map that is injective because the action of G is assumed to be faithfull on thecentral fiber. We may therefore think of g as a subalgebra of C ∞ ( M ). We will doso, and drop ξ from the notation when no confusion can arise. Notice also thatsince the G -action is J t holomorphic, g is a subalgebra of h ( M, J t ) for all t ∈ B .We consider the ideal h ( M, J t ) of Hamiltonian holomorphic vector fields in h ( M, J t ), the space of holomorphic vector fields with a nontrivial zero set [13].Then the ideal of g given by g = g ∩ h ( M, J t ) consists of the vector fields in g ⊂ C ∞ ( M, T M ) that vanish somewhere, and since this properties is independentof the complex structure J t , g is independent of t .We summarize the discussion above into the following proposition: Proposition 4.1.1.
Let M → B be a family of complex deformations of a manifold X of K¨ahler type. Let G be a compact connected Lie group acting holomorphically on M and trivially on B , and let g be its Lie algebra. If B is restricted to a sufficientlysmall neighborhood of the origin, there exists a smooth trivialization M t ≃ ( M, J t ) such that M t is K¨ahler for all t ∈ B , the action of G on M t is independent of t ,the image of the natural embedding g ⊂ C ∞ ( M, T M ) is contained in h ( M, J t ) forall t ∈ B , and g = g ∩ h ( M, J t ) is an ideal of g that is independent of t . Definition 4.1.2 . Let M → B be a family of complex deformations of a manifold X of K¨ahler type, and let G be a compact connected Lie group acting holomorphicallyon M and trivially on B . A smooth trivialisation M t ≃ ( M, J t ) that satisfies theproperties of Proposition 4.1.1 is said to be an adapted trivialization for M → B relative to G .4.2. The equivariant deformation problem.
Let ( M → B, Θ) be a polarizedfamily of complex deformations of a polarized manifold ( X , Ω), and let G be acompact connected Lie group acting holomorphically on M and trivially on B .Since G is connected it acts trivially on the cohomology of the fibers M t , and inparticular on Θ t . Let ( M, J t ) ≃ M t be an adapted trivialization for M → B, Θ)relative to G . With some abuse of notation, we denote by Θ t the polarizationinduced on M via the isomorphism M t ≃ ( M, J t ). Then we have the following: Lemma 4.2.1.
Let ( M → B, Θ) be a polarized deformation of ( X , Ω) ≃ ( M , Θ ) provided with a holomorphic action of a compact connected Lie group G actingtrivially on B . Consider an adapted trivialization ( M, J t ) ≃ M t , and let g be anextremal metric on X that represents the K¨ahler class Ω . Then, if B is restrictedto a sufficiently small neighborhood of the origin, there exists a smooth family of G -invariant K¨ahler metrics g t on ( M, J t ) that represent the K¨ahler classes Θ t ,and such that g is identified to the metric g by the isomorphism X ≃ M up toconjugation by an element of Aut( X ) . Proof.
The connected component of the identity in the group of isometries denotedIsom ( X , g ) is a maximal compact connected subgroup of Aut( X ) [5]. Thus, we mayassume that G ⊂ Isom ( X , g ) up to conjugation by an element of Aut( X ).Let g be the G -invariant metric on ( M, J ) that corresponds to g by the isomor-phism X ≃ ( M, J ). By the Kodaira-Spencer theory, we can extend g to a smoothfamily of K¨ahler metrics g t on ( M, J t ) that represent Θ t for t s sufficiently small.We can average these metrics if necessary to make of them G -invariant. Since G acts isometrically on g , the averaging process leaves g unchanged. On the otherhand, G acts trivially on the cohomology, and hence, on Θ t . So the averaging ofthe metric g t does not change the K¨ahler class that the metric represents. Thisfinishes the proof. (cid:3) Definition 4.2.2 . Let ( M → B, Θ) be a polarized family of deformations of ( X , Ω)provided with a holomorphic action of a compact connected Lie group G actingtrivially on B . Assume that Ω is represented by an extremal metric g . A smoothfamily of G -invariant K¨ahler metrics g t satisfying the properties given by Lemma4.2.1 for an adapted trivialization is said to be an adapted smooth family of K¨ahlermetrics .Let us notice that for the adapted family of metrics g t of Lemma 4.2.1, the metric g coincides with the metric g on X up to the conjugation by an automorphism.Throughout the rest of the paper, we shall assume that the isomorphism X ≃ M has been so conjugated so that g and g coincide.In our considerations below, the group G ′ of § X , g ) of g .4.3. Analytical considerations.
Let ( M → B, Θ) be a polarized deformation of( X , Ω), and let G be a compact Lie group acting holomorphically on M → B andtrivially on B . We assume that G is contained in G ′ . Let g be an extremal metric on X that represents the class Ω. We consider an adapted trivialization M t ≃ ( M, J t )and adapted family of K¨ahler metrics g t on ( M, J t ) (cf. § g = g .Let L k ( M ) be the k th Sobolev space defined by g , and let L k,G be space ofelements in L k that are G -invariant. The latter can be defined as the Banach spacecompletion of C ∞ G ( M ) in the L k norm.We denote by ω t the K¨ahler form of the metric g t on ( M, J t ) of the adaptedfamily. Let H t ( M ) be the space of g t -harmonic real (1 , M, J t ). ByHodge theory, we have that H t ( M ) ≃ H , ( M, J t )) ∩ H ( M, R ) . Since G is connected, it acts trivially on the cohomology of M . By the uniquenessof the harmonic representative of a close form, it therefore acts trivially on H t also. Further, the Kodaira-Spencer theory shows that all the spaces H t ( M ) areisomorphic for t s sufficiently small, and form the fibers of a vector bundle H ( M ) → B . For a given function φ ∈ L k +4 ,G ( M ) and α ∈ H t ( M ) sufficiently small, one candefine a new K¨ahler metric g t,α,φ on ( M, J t ) with K¨ahler form ω t,α,φ = ω t + α + dd c φ. EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 17
By definition, the deformed metric g t,α,φ is automatically invariant under the G -action and represents the K¨ahler class Θ t,α = Θ t + [ α ].The real and complex Lichnerowicz operator of g t,α,φ will be denoted L t,α,φ and L t,α,φ respectively. The space of Killing potentials for the metric g t,α,φ is given by i ker L t,α,φ .Since G acts isometrically on g t,α,φ , the Lie algebra g ⊂ h ( M, J t ) consists ofHamiltonian Killing fields for the metric g t,α,φ as well. Let H g t,α,φ be the space ofholomorphy potentials corresponding to the Killing fields in g for the metric g t,α,φ .The space H g t,α,φ consists of purely imaginary functions of the form iv where v isa Killing potential for some Killing vector field in g relative to the metric g t,α,φ .Using the notations of § H z t,α,φ as the G -invariant part of H g t,α,φ .An essential feature of H z t,α,φ is that it is identified to R ⊕ g G , where g G is the Liesubalgebra of Ad ( G )-invariant vector fields in g (or equivalently the G -invariantone when g is considered as a Lie subalgebra of f ( M )).It follows that the spaces H z t,α,φ have constant dimension and that they are thefibers of a vector bundle H z over a neighborhood of the origin in the total spaceof the bundle L k +4 ,G ( M ) ⊕ H ( M ) → B .The L -norm on L k ′ ,G ( M ) induced by the Riemannian metric g t,α,φ allows us todefine the orthogonal W k ′ ,t,α,φ of i H z t,α,φ and an orthogonal direct sum L k ′ ,G ( M ) = i H z t,α,φ ⊕ W k ′ ,t,α,φ varying smoothly with ( t, α, φ ). This construction provides Banach bundles W k ′ → V where V is a sufficiently small neighborhood of the origin in the total space of L k +4 ,G ( M ) ⊕ H ( M ) → B . We shall denote by π Wt,α,φ : L k ′ ,G ( M ) → W k ′ ,t,α,φ and π Gt,α,φ : L k ′ ,G ( M ) → i H z t,α,φ the canonical projection associated to the above splitting. The reduced scalar cur-vature s Gt,α,φ of g t,α,φ is given by s Gt,α,φ = π Wt,α,φ ( s g t,α,φ ) = (1l − π Gt,α,φ )( s g t,α,φ ). Weare looking for particular extremal metrics near g , namely the one with vanishingreduced scalar curvature(16) s Gt,α,φ = 0 . The LHS of (16) can be interpreted as a section of the bundle W k → V . Our goalis to seek the zeroes of this section of Banach bundle. Keeping a more prosaic stylewe can express equation (16) more concretely by using suitable trivialisations of therelevant bundles as follows: the metric g induces an L -norm and an associatedorthogonal projection P = π W : L k +4 ,G ( M ) → W k, . For parameters ( t, α, φ )sufficiently small, the restriction P : W k,t,α,φ → W k, , is automatically an isomorphism.The bundle of harmonic real (1 , H ( M ) → B , admits a trivialization ina neighborhood of the central fiber. Thus we have a smooth isomorphism of vectorbundle h , up to the cost of shrinking B to some smaller neighborhood of the origin, which commutes with the canonical projections B × H ( M ) h / / % % ❑❑❑❑❑❑❑❑❑❑ H ( M ) | | ②②②②②②②② B and such that h restricted to the central fiber is the identity. We shall use thenotation h ( t, α ) = h t ( α ) ∈ H t ( M ).Let U be a sufficiently small open neighborhood of the origin in B × H ( M ) × W k +4 , such that the following map is defined(17) S : U → B × W k, ( t, α, φ ) (cid:16) t, P (cid:16) s Gt,h t ( α ) ,φ (cid:17)(cid:17) Lemma 4.3.1.
The map S is C and its differential is a Fredholm operator. Assum-ing that the K¨ahler metric g on X has vanishing reduced scalar curvature s Gg = 0 ,the differential at ( t, α, φ ) = 0 is given by given by a linear operator of the form (cid:18)
1l 0 ∗ S G,g (cid:19) where S G,g ( ˙ α, ˙ φ ) = − L g ˙ φ + P ( ˙ s Gg ( ˙ α )) is the differential of P ( s G ) at g in the direction of ( ˙ α, ˙ φ ) . In the case where ˙ α istracefree, we have S G,g ( ˙ α,
0) = ˙ s Gg ( ˙ α ) = P ( G g ( h ˙ α, dd c s g i ) − h ˙ α, ρ i ) . Proof.
The map S is C since the reduced scalar curvature depends in a C mannerof the data ( t, α, φ ). The computation of the differential of S is deduced fromLemma 2.2.7. (cid:3) At this point, we may compute the index of the differential of S : Lemma 4.3.2.
Under the assumption s Gg = 0 , the index of the differential of S atthe origin is equal to h , ( X ) .Proof. By Lemma 4.3.1, the operator S G,g is a compact perturbation of the map H ( M ) × W k +4 , → W k, ( α, φ )
7→ − L g φ The Lichnerowicz operator L g : W k +4 , → W k, has index 0. We conclude that thedifferential of S has index dim H ( M ) = h , ( X ). (cid:3) Surjectivity.
We return to the study of the map S with the notations of § Proposition 4.4.1.
Under the additional assumption that g is an extremal metricon X with s Gg = 0 , the map S defined at (17) is a submersion at the origin if andonly the relative Futaki invariant F cG, Ω is non degenerate at Ω , the K¨ahler class of g . EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 19
Proof.
The cokernel of the differential of S is identified to ψ ∈ W k, such that h L g ˙ φ, ψ i = 0 , h P ( ˙ s Gg ( ˙ α )) , ψ i = 0for all ˙ φ ∈ W k +4 , and ˙ α g -harmonic (1 , M, J ) ≃ X .The first equation implies that L g ψ = 0. Therefore we have ψ ∈ i H z ′ g ≃ R ⊕ z ′ .Let Ξ ∈ z ′ be the Killing field represented by ψ . Using the second condition in theparticular case where ˙ α is tracefree, we see that0 = h P ( ˙ s Gg ( ˙ α )) , ψ i = h ˙ s Gg ( ˙ α ) , ψ i = Z X ψ ˙ s Gg ( ˙ α ) dµ g = ˙ F cG, Ξ , Ω ( ˙ α ) , where the last equality is given by Lemma 2.3.4. The relative Futaki nondegen-eracy condition implies that Ξ ∈ z , in other words ψ ∈ i H z g . By definition ψ isorthogonal to i H z g , hence ψ = 0 and S is a submersion.Conversely, it is easy to check that if the relative Futaki invariant is degenerate,then S is not a submersion. (cid:3) Remark 4.4.2 . Under the Futaki nondegeneracy assumption, we may chooseany linear subspace V ⊂ H ( M ) such that the linearized Futaki invariant inducesan injective map p / g → V ∗ . Then the corresponding restriction of S is still asubmersive map at the origin.In conclusion, we obtain the following theorem: Theorem 4.4.3.
Let ( M → B, Θ) be a polarized family of complex deformations ofa polarized manifold ( X , Ω) . Assume that M → B is endowed with a holomorphicaction of a connected compact Lie group G acting trivially on B and that X admitsa G -invariant extremal metric g with K¨ahler class Ω and such that s Gg = 0 .Given an adapted trivialization M t ≃ ( M, J t ) defined for t sufficiently small, let g t be any adapted smooth family of G -invariant K¨ahler metrics on ( M, J t ) repre-senting Θ t (cf. § and § ).Assume that the relative Futaki invariant F cG, Ω is non degenerate at g , thenchoose a space V ⊂ H ( M ) such that the linearized relative Futaki invariant re-stricted to p / g → V ∗ is injective. Then, the space of solutions S = { ( t, α, φ ) ∈ U | α ∈ V and s Gg t,ht ( α ) ,φ = 0 } is a smooth manifold of real dimension dim V + dim B , in a sufficiently small neigh-borhood of the origin. For any ( t, α, φ ) ∈ S , α and φ are automatically smooth.The canonical projection S → B is a submersion near the origin and the fibersare dim V -dimensional submanifold of S corresponding to families of G -invariantK¨ahler metrics g t,α,φ on M t with vanishing reduced scalar curvature representing aperturbation of the polarization given by Θ t + [ α ] .Proof. The hypothesis imply that the map S restricted to U ′ = { ( t, α, φ ) ∈ U k α ∈ V } is a submersion at the origin. The kernel K of the differential of S has dimensionequal to its index dim V . Let π K : W k +4 → K be the orthogonal projection onto K . By definition, the map U ′ π K × S −→ K × B × W k, is an isomorphism. The implicit function theorem provides the desired solutionsparametrized by B × K . (cid:3) Remark 4.4.4 . When q / g = 0, the Futaki invariant is automatically non degen-erate and Theorem 4.4.3 applies for any subspace V ⊂ H ( M ). Setting V = 0, thetheorem provides for every t sufficiently small a unique extremal metric on M t withK¨ahler class Θ t . In other words, the original polarization Θ does not have to beperturbed in this case.4.5. Generalization to nonconnected groups.
Although to this point the group G has been assumed to be connected, this assumption is not necessary to a large ex-tent. The connectedness was used to ensure that G acts trivially on the cohomologyof the manifold, hence on harmonic forms. Thus, we merely need the assumptionthat G acts trivially on on the relevant K¨ahler classes.For instance, if G is contained in the connected component of the identity ofAut( X ), then G acts trivially on the cohomology of X . We can check that thedefinition of the Futaki invariant relative to G still makes sense. The analysisdevelopped at § G -equivariantly. Theonly difference in this more general framework will appear in Proposition 4.4.1. Forin order to make sure that this proposition still holds, we should change slightly thedefinition of relative Futaki nondegeneracy. Let us recall that the linearized relativeFutaki invariant induces a map z ′ / z → ( H , ( X ) ∩ H ( X , R )) ∗ . If G is connected, z ′ agrees with the space of G -invariant Hamiltonian Killing fields on ( X , g ). But if G is not connected, there is a residual action of G on z ′ by a finite group action.The space of G -invariant vector fields in z ′ is denoted z ′ ,G . Similarly we denote by z ,G the G -invariant part of z . There is an embedding z ′ ,G / z ,G ⊂ z ′ / z and weshall say that the Futaki invariant F c Ω ,G is nondegenerate at g if the map z ′ ,G / z ,G → ( H , ( X ) ∩ H ( X , R )) ∗ is injective.Using this new definition of the Futaki nondegeneracy, we can drop the connec-tivity assumption on G and just assume that G ⊂ G ′ = Isom ( X , g ) in Theorem4.4.3. We can then derive the same conclusions.5. Applications
In this section we apply Theorem A in some particular situations, and especiallyto produce new examples of K¨ahler manifolds with extremal metrics.5.1.
Relation to LeBrun-Simanca deformation theory.
It is easy to see thatTheorem A enables us to recover the deformation theory of [14].5.1.1.
Case G = { } . Let g be a K¨ahler metric on X with K¨ahler class Ω. Then,we have s Gg = s g − ¯ s g , where ¯ s g is the average of s g on X . So the condition s Gg = 0is equivalent to the property that g has constant scalar curvature. Furthermore theFutaki invariant relative to G = { } agrees with the non-relative Futaki invariant.In this case, Theorem A is equivalent to the deformation theory for cscK metricsof [14].5.1.2. Case G = G ′ . If g is extremal, then G ′ = Isom ( X , g ) is a maximal connectedcompact subgroup of Aut( X ). Then every extremal metric is G -invariant, uptoconjugation, and the condition s Gg = 0 for a G -invariant K¨ahler metric is equivalentto the metric being extremal. EFORMATION OF EXTREMAL METRICS, COMPLEX MANIFOLDS 21
Theorem A applies. In the case of trivial complex deformations with B = { } ,one recovers the openness theorem of [14]. We actually seem to have a more generalresult for it is possible to allow complex deformations with a holomorphic actionof G acting trivially on B .5.2. Deformations with maximal torus symmetry.
The deformation theoryunder a maximal compact torus symmetry is particularly well behaved. Let X be complex manifold and g an extremal metric on X with K¨ahler class Ω. Weshall denote by G = T n ⊂ Aut( X ) a maximal compact torus. Up to conjugationby an automorphism, we may assume that the metric g is G -invariant so that G ⊂ G ′ = Isom ( X , g ). The vector field Ξ = J grad s g is in z ′ ⊂ z ′′ . But in thiscase z ′′ = z by maximality of z (cf. proof of Lemma 2.3.7). Hence Ξ ∈ z ⊂ g ,which implies s Gg = 0. The Futaki invariant is trivial, hence nondegenerate (cf.Lemmas 2.3.7). So we may apply Theorem A to polarized deformations ( M → B, Θ)of ( X , Ω) endowed with a holomorphic action of G acting trivially on B . Thus wehave proved Corollary B.5.3. The Mukai-Umemura -fold. Let X Π be the Mukai-Umemura 3-fold of § M → C of X Π that we described there, which isprovided with an holomorphic action of a group G isomorphic to the dihedral groupof order 8 or the semi-direct product of Z / ⋊ S . Proofs of Corollaries C . We use the fact that X Π admits a K¨ahler-Einstein metric[7]. Notice that although the group G is disconnected, it acts trivially on thecohomology of M t (cf. § X Π at the K¨ahler-Einstein metric is nondegenerate in the sense § G -invariant holomorphic vector fields on X Π is reduced to 0 since the action of G does not fix any point in P . Thus z ′ ,G = 0 and the Futaki invariant is nondegenerate necessarily nondegenerate. Wemay apply Theorem 4.4.3, with the choice of space V = 0. Thus every class Θ t is represented by a G -invariant extremal K¨ahler metric g t for t sufficiently small.Thus the holomorphic vector field grad s g t must be G invariant. As the action of G does not fix any point in P , it follows that there are no nontrivial G -invariantholomorphic vector fields on M t . Thus s g t is constant and this finishes the proof ofCorollary C. (cid:3) References [1] V. Apostolov, D.M.J. Calderbank, P. Gauduchon & C.W. Tøennesen-Friedman,
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