aa r X i v : . [ m a t h . DG ] A ug RICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONALHOLONOMY
JOHANNES NORDSTR ¨OM
Abstract.
Let G be one of the Ricci-flat holonomy groups SU ( n ), Sp ( n ), Spin (7) or G , and M a compact manifold of dimension 2 n , 4 n , 8 or 7, respectively. We prove that the naturalmap from the moduli space of torsion-free G -structures on M to the moduli space of Ricci-flatmetrics is open, and that the image is a smooth manifold. For the exceptional cases G = Spin (7)and G we extend the result to asymptotically cylindrical manifolds. Introduction
The possible holonomy groups of simply-connected non-symmetric irreducible Riemannian man-ifolds were classified by Berger [1]. ‘Berger’s list’ contains several infinite families, and the twoexceptional cases
Spin (7) and G , appearing as the holonomy of manifolds of dimension 8 and7 respectively. In many cases, an effective approach to studying G -metrics (by which we meanmetrics with holonomy contained in G ) is to define them in terms of certain closed differentialforms, equivalent to torsion-free G -structures . A G -structure defines a Riemannian metric, andif its torsion vanishes (which is a first-order differential equation) then the induced metric hasholonomy contained in G . For G = SU ( n ), Sp ( n ), Spin (7) or G we define a G -manifold to be aconnected oriented manifold of dimension 2 n , 4 n , 8 or 7 respectively, equipped with a torsion-free G -structure and the associated Riemannian metric. G -metrics are Ricci-flat for G = SU ( n ), Sp ( n ), Spin (7) or G . For compact manifolds M.Y.Wang [22, Theorem 3.1] proved a local converse: any small Ricci-flat deformation of a G -metricstill has holonomy contained in G . In other words, the moduli space W G of G -metrics is an opensubset of the moduli space W of Ricci-flat metrics. This is an analogue of a result of Koiso [10]on deformations of K¨ahler-Einstein metrics. Wang proves the result case by case, but asks if thereis a general proof.In this paper we observe that the problem can be reduced in a uniform way to showing unob-structedness for deformations of torsion-free G -structures. This has in turn been given a uniformtreatment by Goto [7]. As part of the proof we provide a clear summary of the deformation theoryof Ricci-flat metrics on a compact manifold (a special case of deformation theory for Einsteinmetrics used by Koiso [10]). This treatment makes it easier to extend the results to other types ofcomplete manifolds, and we will discuss the asymptotically cylindrical case in some detail.If M is a compact G -manifold then the group D of diffeomorphisms of M isotopic to the identityacts on the space of torsion-free G -structures by pull-backs. The resulting quotient is the modulispace M G of torsion-free G -structures on M , and is known to be a manifold. This is due to Tian[19] and Todorov [20] in the Calabi-Yau ( G = SU ( n )) case, and Joyce in the exceptional cases (see[9, § .
4, 10 . D also acts on the space of Riemannian metrics, and we let W G and W denotethe moduli spaces of G -metrics and Ricci-flat metrics respectively. In § Theorem I.
Let G = SU ( n ) , Sp ( n ) , Spin (7) or G , and let M be a compact G -manifold. Then W G is open in W . Moreover, W G is a smooth manifold and the natural map m : M G → W G that sends a torsion-free G -structure to the metric it defines is a submersion. Mathematics Subject Classification.
Remark . It is easy to see that W G is also closed in W , so it is a union of connected components.It seems to be an open problem whether there exist any compact Ricci-flat manifolds without aholonomy reduction. Remark . The quotient of the space of G -metrics by the group of all diffeomorphisms of M (not just the ones isotopic to the identity) is a quotient of W G with discrete fibres and in generalan orbifold (cf. remark 3.11).The case G = G of theorem I was proved by M.Y. Wang [22, Theorem 3 . G = Sp ( n )or Spin (7), Wang showed that W G ⊆ W is open (so the statement of theorem I is stronger).Manifolds with holonomy in SU ( n ) are Calabi-Yau manifolds, i.e. Ricci-flat K¨ahler manifolds.The case G = SU ( n ) of theorem I is therefore a special case of a more general result by Koiso onEinstein deformations of K¨ahler-Einstein metrics.Let X n be a compact K¨ahler-Einstein manifold. Koiso [10, Theorem 0 .
7] shows that if theEinstein constant e (equivalently the first Chern class c ( X )) is non-positive and the complexdeformations of X are unobstructed, then any small Einstein deformation of the metric is K¨ahlerwith respect to some perturbed complex structure. In other words, the map from the moduli spaceof K¨ahler-Einstein structures to the moduli space of Einstein metrics is open (see e.g. [2, § G = SU ( n ) follows from Koiso’s theorem, except for theclaim that W SU ( n ) is smooth (and not just an orbifold). Remark . Dai, X. Wang and Wei [5] use the fact that W G is open in W to deduce that anyscalar-flat deformation of a Ricci-flat G -metric on a compact manifold remains a G -metric.The proof of theorem I given in § G = G .First, we observe that the point-wise surjectivity of the derivative of m follows from a well-knownproperty of Laplacians on manifolds with reduced holonomy noted by Chern [4]. This makes iteasy to see that the proof applies also for the other Ricci-flat holonomy groups, provided thatthe deformations of torsion-free G -structures are unobstructed. Second, we streamline some partsof the deformation theory for Ricci-flat metrics. This makes it easier to generalise the result tocertain non-compact settings.One relevant type of complete non-compact Riemannian manifolds are exponentially asymptot-ically cylindrical (EAC) ones (defined in § § G = Spin (7) or G ) torsion-free G -structurescarries over to the EAC case, so that there are smooth moduli spaces M G and W of torsion-freeEAC G -structures and Ricci-flat EAC metrics on an EAC G -manifold M . Theorem I ′ . Let G = Spin (7) or G , and M an EAC G -manifold. Then W G is open in W .Moreover, W G is a smooth manifold and the natural map m : M G → W G is a submersion. In [11] Kovalev proves the analogous result for EAC Calabi-Yau manifolds, by an extensionof Koiso’s arguments for the compact K¨ahler-Einstein case. The discussion in subsection 3.5 ofdeformations of EAC Ricci-flat metrics is similar to that in [11], while the necessary results ondeformations of EAC G -structures are taken from [17]. Remark . One may consider the structure of the map m in greater detail. For the exceptionalcases G = G and Spin (7), one can use the characterisation of torsion-free G -structures in termsof parallel spinors (cf. M.Y. Wang [21]) to show that m is a diffeomorphism if the holonomy of M is exactly G for any G -metric (this depends only on the topology of M ), and that M G is ingeneral a disjoint union of fibre bundles over W G with real projective plane fibres (the componentscorrspond to different spin structures on M ). See [16, § . ICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONAL HOLONOMY 3 is a locally trivial fibration with compact fibres over the moduli space of Calabi-Yau metrics (butdoes not describe the fibres further).
Acknowledgements.
I am grateful to Alexei Kovalev for many helpful discussions.2.
Preliminaries
We describe how a metric with holonomy G = Spin (7), G , SU ( n ) or Sp ( n ) can be defined interms of a torsion-free G -structure. This is a set of differential forms, that are both parallel andharmonic. For more background on manifolds with special holonomy see e.g. Joyce [9] or Salamon[18].2.1. Holonomy.
We define the holonomy group of a Riemannian manifold. For a fuller discussionof holonomy see e.g. [9, Chapter 2].
Definition 2.1.
Let M n be a manifold with a Riemannian metric g . If x ∈ M and γ is a closedpiecewise C loop in M based at x then the parallel transport around γ (with respect to theLevi-Civita connection) defines an orthogonal linear map P γ : T x M → T x M . The holonomy group Hol ( g, x ) ⊆ O ( T x M ) at x is the group generated by { P γ : γ is a closed loop based at x } .If x, y ∈ M and τ is a path from x to y we can define a group isomorphism Hol ( g, x ) → Hol ( g, y )by P γ P τ ◦ P γ ◦ P − τ . Provided that M is connected we can therefore identify Hol ( g, x ) witha subgroup of O ( n ), independently of x up to conjugacy, and talk simply of the holonomy groupof g .There is a correspondence between tensors fixed by the holonomy group and parallel tensorfields on the manifold. Proposition 2.2 ([9, Proposition 2 . . . Let M n be a Riemannian manifold, x ∈ M and E avector bundle on M associated to T M . If s is a parallel section of E then s ( x ) is preserved by Hol ( g, x ) . Conversely if s ∈ E x is preserved by Hol ( g, x ) then there is a parallel section s of E such that s ( x ) = s . Definition 2.3.
Let M n a manifold and G ⊆ O ( n ) a closed subgroup. A G -metric on M is ametric with holonomy contained in G .2.2. Spin (7) -structures.
The stabiliser in GL ( R ) of ψ = dx + dx + dx + dx − dx − dx − dx − dx − dx − dx + dx + dx + dx + dx ∈ Λ ( R ) ∗ (1)is Spin (7) (identified with a subgroup of SO (8) by the spin representation). For an oriented vectorspace V of dimension 8 let Λ Spin (7) V ∗ ⊂ Λ V ∗ be the subset of forms equivalent to ψ under someoriented linear isomorphism V ∼ = R . A Spin (7) -structure on an oriented manifold M is a sectionof the subbundle Λ Spin (7) T ∗ M ⊂ Λ T ∗ M . Since Spin (7) ⊂ SO (8) a Spin (7)-structure ψ naturallydefines a Riemannian metric g ψ on M . Note that ψ is self-dual with respect to this metric.We make a note of the decomposition of Λ R into irreducible representations of Spin (7).Firstly it splits into the self-dual and anti-self-dual parts Λ ± R . We let Λ d R denote an irreduciblecomponent of rank d . Then Λ R = Λ R ⊕ Λ R ⊕ Λ R , (2a)Λ − R = Λ R . (2b)The tangent space at ψ to the space of Spin (7)-structures Γ(Λ
Spin (7) T ∗ M ) is Γ( E ψ ), where E ψ ⊂ Λ T ∗ M is a Spin (7)-invariant linear subbundle. More precisely, the
Spin (7)-structure ψ determines a decomposition of Λ T ∗ M modelled on (2), and E ψ = Λ ⊕ ⊕ T ∗ M .A Spin (7)-structure ψ is torsion-free if it is parallel with respect to the metric it induces.It follows immediately from proposition 2.2 that a metric g on M has holonomy contained in Spin (7) if and only if it is induced by a torsion-free
Spin (7)-structures.
JOHANNES NORDSTR ¨OM
The condition that
Hol ( g ) ⊆ Spin (7) imposes algebraic constraints on the curvature of g . Inparticular any Spin (7)-metric is Ricci-flat (see [18, Corollary 12 . ψ can usefully be rewritten as dψ = 0 (see [18, Lemma 12 . G -structures. Recall that G can be defined as the automorphism group of the normedalgebra of octonions. Equivalently, G is the stabiliser in GL ( R ) of ϕ = dx + dx + dx + dx − dx − dx − dx ∈ Λ ( R ) ∗ . (3)For an oriented vector space V of dimension 7 let Λ G V ∗ ⊂ Λ V ∗ be the subset consisting offorms equivalent to ϕ under some oriented linear isomorphism V ∼ = R . A G -structure on anoriented manifold M is a section ϕ of the subbundle Λ G T ∗ M ⊂ Λ T ∗ M , and naturally definesa Riemannian metric g ϕ on M .The typical fibre of Λ G T ∗ M is isomorphic to GL ( R ) /G , so by dimension-counting Λ G T ∗ M is an open subbundle of Λ T ∗ M . Thus the tangent space at ϕ to the space of G -structuresΓ(Λ G T ∗ M ) is Ω ( M ) = Γ( E ϕ ), if we let E ϕ = Λ T ∗ M .A G -structure ϕ is torsion-free if it is parallel with respect to the metric it induces. A metric g on M has holonomy contained in G if and only if it is induced by a torsion-free G -structure. G -metrics are Ricci-flat (see [18, Proposition 11 . G -structure ϕ istorsion-free if and only if dϕ = 0 and d ∗ ϕ ϕ = 0 (where the codifferential d ∗ ϕ is defined using themetric induced by ϕ , see [18, Lemma 11 . SU ( n ) -structures. Let z k = x k − + ix k be complex coordinates on R n . Then the stabiliserin GL ( R n ) of the pair of formsΩ = dz ∧ · · · ∧ dz n ∈ Λ n ( R n ) ∗ ⊗ C (4a) ω = i ( dz ∧ d ¯ z + · · · dz n ∧ d ¯ z n ) ∈ Λ ( R n ) ∗ (4b)is SU ( n ). For an oriented real vector space V of dimension 2 n let Λ SU ( n ) V ∗ ⊂ Λ n V ∗ C ⊕ Λ V ∗ bethe subset of pairs (Ω , ω ) equivalent to (Ω , ω ) under some oriented isomorphism V ∼ = R n . An SU ( n ) -structure on an oriented manifold M n is a section (Ω , ω ) of the subbundle Λ SU ( n ) T ∗ M ⊂ Λ n T ∗ C M ⊕ Λ T ∗ M . It naturally defines an almost complex structure and a Riemannian metricon M , such that Ω has type ( n, V is given by both ( − n ( n − ( i ) n Ω ∧ ¯Ωand n ! ω n (cf. Hitchin [8, § SU ( n )-structure is torsion-free if it is parallel with respect to the metric it induces, anda metric on M n has holonomy contained in SU ( n ) if and only if it is induced by a torsion-free SU ( n )-structure.(Ω , ω ) is torsion-free if and only if d Ω = dω = 0. Then the induced almost complex structure isintegrable, the Riemannian metric is a Ricci-flat K¨ahler metric, and Ω is a holomorphic ( n, M n equipped with a torsion-free SU ( n )-structure is called an SU ( n ) -manifold or Calabi-Yau n -fold .2.5. Sp ( n ) -structures. Let q k = x k − + ix k − + jx k − + kx k be quaternionic coordinateson R n . Then we may write dq ∧ d ¯ q + · · · + dq n ∧ d ¯ q n = − iω I + jω J + kω K ) , with ω I , ω J , ω K ∈ Λ ( R n ) ∗ . The stabiliser in GL ( R n ) of this triple of 2-forms is Sp ( n ). For anoriented real vector space V of dimension 4 n let Λ Sp ( n ) V ∗ ⊂ (Λ V ∗ ) ⊗ be the subset of triples( ω I , ω J , ω K ) equivalent to ( ω I , ω J , ω K ) under some oriented isomorphism V ∼ = R n . An Sp ( n ) -structure on an oriented manifold M n is a section of the subbundle Λ Sp ( n ) T ∗ M ⊂ (Λ T ∗ M ) ⊗ .It is torsion-free if it is parallel with respect to the induced metric, and a metric on M n hasholonomy contained in Sp ( n ) if and only if it is induced by a torsion-free Sp ( n )-structure.Equivalently, ( ω I , ω J , ω K ) is torsion-free if and only if dω I = dω J = dω K = 0. Then the metricof M is Ricci-flat, and M has a triple I, J, K of anti-commuting integrable complex structures,such that ω I is the K¨ahler form and ω J + iω K a holomorphic (2 , I , etc. M n equipped with a torsion-free Sp ( n )-structure is called an Sp ( n ) -manifold or hyperK¨ahler manifold . ICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONAL HOLONOMY 5
Laplacians.
For a Riemannian manifold with holonomy H one can define a LichnerowiczLaplacian on vector bundles associated to the H -structure. On differential forms this agrees withthe usual Hodge Laplacian, as is explained in Besse [2, § M n has holonomy group Hol ( M ) ⊆ H (where H is a closedsubgroup of O ( n )), and a corresponding H -structure. Let ρ : H → GL ( E ) be a representationof H , and E ρ the corresponding associated vector bundle. Let h ad be the vector bundle inducedby the adjoint representation. h ad can be identified with a subbundle of Λ T ∗ M , and because Hol ( M ) ⊆ H the Riemannian curvature tensor R is a (symmetric) section of h ad ⊗ h ad . We usethe Lie algebra representation Dρ : h → End ( E ) to define( Dρ ) : h ⊗ h → End ( E ) , a ⊗ b Dρ ( a ) ◦ Dρ ( b ) . This induces a bundle map h ad ⊗ h ad → End ( E ρ ). The symmetry of R implies that ( Dρ ) ( R ) is aself-adjoint section of End ( E ρ ). Definition 2.4.
Let M be a Riemannian manifold with Hol ( M ) ⊆ H and ρ a representation of H .The Lichnerowicz Laplacian on the associated vector bundle E ρ is the elliptic formally self-adjointoperator △ ρ = ∇ ∗ ∇ − Dρ ) ( R ) : Γ( E ρ ) → Γ( E ρ ) , where ∇ is the connection on E ρ induced by the Levi-Civita connection on M . Lemma 2.5.
Let M n be a Riemannian manifold. The Lichnerowicz Laplacian corresponding tothe standard representation of O ( n ) on Λ m ( R n ) ∗ is the usual Hodge Laplacian △ on Λ m T ∗ M .Proof. See [2, Equation (1 . (cid:3) Lemma 2.6 (cf. [9, Theorem 3 . . . Let M be a Riemannian manifold with Hol ( M ) ⊆ H and φ : E → F an equivariant map of H -representations ( E, ρ ) , ( F, σ ) . φ induces a bundle map E ρ → F σ , and the diagram below commutes. Γ( E ρ ) φ ✲ Γ( F σ )Γ( E ρ ) △ ρ ❄ φ ✲ Γ( F σ ) △ σ ❄ In particular, if ρ , ρ are H -representations then △ ρ ⊕ ρ = △ ρ ⊕ △ ρ .Proof. Clear from the fact that the Lichnerowicz Laplacian is defined naturally by the represen-tations. (cid:3)
Suppose that Λ m ( R n ) ∗ splits as a direct sum of representations of H . On a manifold M with ho-lonomy contained in H there is a corresponding splitting of Λ m T ∗ M into H -invariant subbundles.Lemma 2.6 implies that the Hodge Laplacian commutes with the projections to the subbundles.Hence there is also a decomposition for the harmonic forms (see [9, Theorem 3 . . Asymptotically cylindrical manifolds.
A non-compact manifold M is said to have cylin-drical ends if M is written as union of two pieces M and M ∞ with common boundary X , where M is compact, and M ∞ is identified with X × R + by a diffeomorphism (identifying ∂M ∞ with X × { } ). X is called the cross-section of M . Let t be a smooth real function on M which is the R + -coordinate on M ∞ , and negative on the interior of M . A tensor field s on M is said to be expo-nentially asymptotic with rate δ > s ∞ on M if e δt k∇ k ( s − s ∞ ) k is bounded on M ∞ for all k ≥
0, with respect to a norm defined by an arbitrary Riemannianmetric on X . JOHANNES NORDSTR ¨OM
A metric on M is called EAC (exponentially asymptotically cylindrical) if it is exponentiallyasymptotic to a product metric on X × R . Similarly a G -structure is said to be EAC if it is exponen-tially asymptotic to a translation-invariant G -structure on X × R which defines a product metric.A diffeomorphism φ of M is called EAC if it is exponentially close to a product diffeomorphism( x, t ) (Ξ( x ) , t + h ) of X × R in a similar exponential sense. Remark . If an EAC metric has reduced holonomy then so does the induced metric on the cross-section. In particular, the cross-section of an EAC
Spin (7)-manifold is a compact G -manifold,and the cross-section of an EAC G -manifold is a Calabi-Yau 3-fold.On an asymptotically cylindrical manifold M it is useful to introduce weighted H¨older norms .Let E be a vector bundle on M associated to the tangent bundle, k ≥ α ∈ (0 ,
1) and δ ∈ R . Wedefine the C k,αδ -norm of a section s of E in terms of the usual H¨older norm by k s k C k,αδ = k e δt s k C k,α . (5)Denote the space of sections of E with finite C k,αδ -norm by C k,αδ ( E ). Up to Lipschitz equivalencethe weighted norms are independent of the choice of asymptotically cylindrical metric, and of thechoice of t on the compact piece M . In particular, the topological vector spaces C k,αδ ( E ) areindependent of these choices.The main importance of the weighted norms is that elliptic asymptotically translation-invariantoperators acts as Fredholm operators on the weighted spaces of sections. In particular, this appliesto the Hodge Laplacian of an EAC metric. Theorem 2.8.
Let M be an asymptotically cylindrical manifold. If δ > with δ smaller thanany positive eigenvalue of the Laplacian on X then △ : C k +2 ,α ± δ (Λ m T ∗ M ) → C k,α ± δ (Λ m T ∗ M ) (6) is Fredholm for all m . The index of (6) is ∓ ( b m − ( X ) + b m ( X )) .Proof. The Fredholm result is a special case of Lockhart and McOwen [13, Theorem 6.2], whilethe index formula can be found in Lockhart [12, §
3] (or Melrose [14, § (cid:3) This can be used to deduce results analogous to Hodge theory for compact manifolds. Let H m denote the space of bounded harmonic m -forms on M , and H m ∞ the translation-invariantharmonic forms on X × R . H m ∞ = H mX ⊕ dt ∧ H m − X , where H mX are the harmonic forms on X . Any φ ∈ H m is asymptotically translation-invariant; let B ( φ ) ∈ H m ∞ denote its limit. We can write B ( φ ) = B a ( φ ) + dt ∧ B e ( φ ) ∈ H mX ⊕ dt ∧ H m − X . Then H m = H mabs ⊕ H mE , where H mabs is the kernel of B e : H m → H m − X , and H mE ⊂ H m is the subspace of exact forms. Theorem 2.9.
Let M be an EAC manifold. The natural map H mabs → H m ( M ) is an isomorphism. Dually H mE is isomorphic to the kernel of the homomorphism e : H mcpt ( M ) → H m ( M ) inducedby the natural chain inclusion Ω ∗ cpt ( M ) → Ω ∗ ( M ). If M has a single end (i.e. the cross-section X is connected) then the long exact sequence for relative cohomology of ( M, X ) shows that e : H cpt ( M ) → H ( M ) is injective. Hence Corollary 2.10.
Let M n be an asymptotically cylindrical manifold which has a single end (i.e.the cross-section X is connected). Then H E = 0 , and H → H ( M ) is an isomorphism. Ricci-flat deformations of G -metrics Deformations of G -metrics. Let G be one of the Ricci-flat holonomy groups SU ( n ), Sp ( n ), Spin (7) or G , and M a compact G -manifold. We explained in § G -metric on a manifold M of the appropriate dimension can be defined in terms of a G -structure, i.e. a section of a subbundleΛ G T ∗ M ⊂ Λ ∗ T ∗ M , which is torsion-free and in particular closed. In order to prove theorem Iwe will use that deformations of G -structures are unobstructed, and the existence of pre-modulispaces . ICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONAL HOLONOMY 7
The tangent space to Γ(Λ G T ∗ M ) at a G -structure χ consists of the sections of the bundle ofpoint-wise tangents to Λ G T ∗ M at χ , which is a vector bundle E χ ⊆ Λ ∗ T ∗ M associated to the G -structure. E χ is a bundle of forms, so the Hodge Laplacian acts on Γ( E χ ). When χ is torsion-freethis is the same as the Lichnerowicz Laplacian from § D of diffeomorphisms of M isotopic to the identity acts on the space of torsion-free G -structures by pull-backs and the quotient is the moduli space M G of torsion-free G -structures.Goto [7] proves that the deformations of torsion-free G -structures are unobstructed in the followingsense: Proposition 3.1.
Let G = SU ( n ) , Sp ( n ) , Spin (7) or G , M a compact G -manifold, and χ atorsion-free G -structure on M . Then there is a submanifold R of the space of C G -structuressuch that (i) the elements of R are smooth torsion-free G -structures, (ii) the tangent space to R at χ is the space of harmonic sections of E χ , (iii) the natural map R → M G is a homeomorphism onto a neighbourhood of χ D in M G . The spaces R are pre-moduli spaces of torsion-free G -structures and can be used as coordinatecharts for M G , which is thus a smooth manifold. The pre-moduli space R near χ can be chosento be invariant under the stabiliser χ . In fact Proposition 3.2.
Let χ ∈ X , and let I χ ⊆ D be the stabiliser of χ . If R is I χ -invariant andsmall enough then I x acts trivially on R and I χ ′ = I x for all χ ′ ∈ R .Proof. Because the tangent space to R consists of harmonic forms, a neighbourhood of χ can beimmersed in (a direct sum of copies of) the de Rham cohomology of M . Because elements of I χ act trivially on cohomology they must fix such a neighbourhood. The reverse inclusion I χ ′ ⊆ I χ follows from [6, Theorem 7 . (cid:3) Killing vector fields.
Before we discuss the deformations of Ricci-flat metrics we makesome remarks about
Killing vector fields . These are the infinitesimal isometries of a Riemannianmanifold (
M, g ), i.e. vector fields V such that the Lie derivative L V g vanishes. Definition 3.3.
Given a metric g on M let δ ∗ : Ω ( M ) → Γ( S ( T ∗ M )) be the symmetric part ofthe Levi-Civita connection ∇ : Ω ( M ) → Γ( T ∗ M ⊗ T ∗ M ).The formal adjoint δ of δ ∗ is the restriction of ∇ ∗ : Γ( T ∗ M ⊗ T ∗ M ) → Ω ( M ) to the symmetricpart Γ( S ( T ∗ M )). Proposition 3.4 ([2, Lemma 1 . . Let g be a Riemannian metric on a manifold M and V avector field. Then L V g = 2 δ ∗ V ♭ , where V ♭ denotes the -form g ( V, · ) . The second Bianchi identity implies that(2 δ + d tr) Ric = 0 (7)for any Riemannian metric. The operator 2 δ + d tr is sometimes called the Bianchi operator, andit also satisfies the following useful identity. Lemma 3.5 ([11, Equation (14)]) . If ( M, g ) is a Ricci-flat manifold then (2 δ + d tr) δ ∗ = △ . Proof.
The anti-symmetric part of ∇ on Ω ( M ) is d , so δ ∗ = ∇ − d . Also tr δ ∗ = d ∗ on Ω ( M ).Using the Weitzenb¨ock formula △ = ∇ ∗ ∇ − Ric we obtain(2 δ + d tr) δ ∗ = 2 ∇ ∗ ∇ − ∇ ∗ d + d tr δ ∗ = 2 ∇ ∗ ∇ − d ∗ d − dd ∗ = △ . (cid:3) Proposition 3.6.
Let ( M, g ) be a Ricci-flat manifold. If V is a Killing field then the -form V ♭ is harmonic. If M is compact then the converse also holds.Proof. δ ∗ V ♭ = 0 ⇒ △ V ♭ = 0 by lemma 3.5. Trivially ∇ V ♭ = 0 ⇒ δ ∗ V ♭ = 0, and if M is compactthen △ V ♭ = ∇ ∗ ∇ V ♭ = 0 ⇒ ∇ V ♭ = 0 by integration by parts. (cid:3) This implies that, for any of the Ricci-flat holonomy groups G , the space of infinitesimal auto-morphisms of a compact G -manifold is ( H ) ♯ . JOHANNES NORDSTR ¨OM
Deformations of Ricci-flat metrics.
We summarise some deformation theory for Ricci-flat metrics. This is essentially taken from the explanation of the deformation theory for Einsteinmetrics in [2, § M n be a compact manifold. The diffeomorphism group D acts on the space of Ricci-flatmetrics on M by pull-backs. We define the moduli space W of Ricci-flat metrics to be the quotientof the space of Ricci-flat metrics by D . (We do not divide by the rescaling action of R + too, as isdone in [2].)Take k ≥
2, and let g be a Ricci-flat Riemannian metric on M . In order to study a neighbourhoodof g D in W we use the usual technique of considering a transverse slice for the diffeomorphismaction. Such a slice argument is explained very carefully in Ebin [6]. In the current setting itis, however, possible to use elliptic regularity to avoid some of the technical subtleties of Ebin’sargument. As in [17, § C k,α ( S T ∗ M ), andlet D k +1 be the C k +1 ,α completion of D ( D k +1 is generated by exp of C k +1 ,α vector fields). Byproposition 3.4 the tangent space to the D k +1 -orbit at g is δ ∗ g C k +1 ,α (Λ ). Let K be the kernel of2 δ g + d tr g in C k,α ( S T ∗ M ). Because g is Ricci-flat, harmonic 1-forms are parallel and therefore L -orthogonal to the image of 2 δ g + d tr g . It follows from lemma 3.5 and the Fredholm alternativefor △ g on Ω ( M ) that there is a direct sum decomposition C k,α ( S T ∗ M ) = δ ∗ g C k +1 ,α (Λ ) ⊕ K. We use a neighbourhood S of g in K as a slice for the D -action. Remark . This is not exactly the same choice of slice as in [2]. It has been used before byBiquard [3] and Kovalev [11].Let Q be the space of Ricci-flat (not a priori smooth) metrics in S – this is the pre-modulispace of Ricci-flat metrics near g . The linearisation of the Ricci curvature functional at a Ricci-flatmetric is given by (cf. [2, Equation (12 . ′ )])( DRic ) g h = △ L h + δ ∗ g (2 δ g + d tr g ) h, (8)where △ L denotes the Lichnerowicz Laplacian on S T ∗ M in the sense of definition 2.4. In par-ticular, on the tangent space K to the slice the linearisation reduces to △ L . This is elliptic soits kernel has finite dimension. Moreover, the kernel of △ L is contained in K : differentiating theBianchi identity (7) at the Ricci-flat metric g gives(2 δ g + d tr g )( DRic ) g = 0 , and hence △ L h = 0 ⇒ △ (2 δ g + d tr g ) h = 0 ⇒ (2 δ g + d tr g ) h = 0 . Definition 3.8.
The space of infinitesimal Ricci-flat deformations of g is the kernel ε ( g ) of △ L in Γ( S ( T ∗ M )).If h ∈ Γ( S T ∗ M ) is tangent to a curve of Ricci-flat metrics in the slice S then of course h ∈ ε ( g ).The converse is not true; in general there may be elements in ε ( g ) which are not tangent to anycurve of Ricci-flat metrics. Thus Q need not be a manifold with tangent space ε ( g ).The image of DRic g is the L -orthogonal complement K ′ to ε ( g ) in K . Let P g be the L -orthog-onal projection to K ′ . The Ricci curvature functional is real analytic. We can apply the implicitfunction theorem to the composition F : S → K ′ : h P g Ric ( h ) (9)to deduce that there is a real analytic submanifold Z ⊆ S whose tangent space at g is precisely ε ( g ) and which contains Q as a real analytic subset. The analyticity implies that if every elementof ε ( g ) is tangent to a curve of Ricci-flat metrics then in fact Q contains a neighbourhood of g in Z . Thus the pre-moduli space Q is a manifold in this case (cf. [10, Corollary 3 . ICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONAL HOLONOMY 9
Note that since K is invariant under the isometry group I g of g we may take S , Z and Q to beinvariant too. An analogue of proposition 3.2 holds. Proposition 3.9.
For any g ′ ∈ Q sufficiently close to g , I g ′ ⊆ I g . Moreover, the identity compo-nents of I g ′ and I g are equal.Proof. The inclusion I g ′ ⊆ I g follows from [6, Theorem 7 . b ( M ), so if I g ′ ⊆ I g then the identitycomponents must be equal. (cid:3) The elements of Z are smooth by elliptic regularity (since the linear part of the equation F ( h ) = 0 defining Z is △ L h = 0), and when Q = Z it is relatively straight-forward to deducefrom the submersion theorem that Q → W is open. In general one needs to do a little bit of extrawork. Theorem 3.10.
Let M be a compact manifold and g a Ricci-flat metric on M . Let Q be thepre-moduli space of Ricci-flat metrics near g , and I g the stabiliser of g in D . Then Q / I g ishomeomorphic to a neighbourhood of g D in W . In particular, if every element of ε ( g ) is integrablethen W is an orbifold near g D .Proof. We wish to extend (9) to a function on a neighbourhood U of g in C k,α ( S T ∗ M ) suchthat F − (0) is a manifold containing the Ricci-flat metrics in U and ensure that Z D k +1 ∩ U ⊆ F − (0). Then we apply the submersion theorem to deduce that Z contains representatives for alldiffeomorphism classes in F − (0) close to g .By the inverse function theorem, any element of a small neighbourhood U of g can be writtenas k + φ ∗ g ′ , with k ∈ K ′ , φ ∈ D k +1 and g ′ ∈ Z . Using proposition 3.9, f : U → g D k +1 , k + φ ∗ g ′ φ ∗ g is a well-defined smooth function. If f ( h ) = φ ∗ g then P f ( h ) is a projection to φ ∗ K ′ , and we cantake F : U → K ′ , h P g P f ( h ) Ric ( h ) . (10)Then DF g maps K ′ onto itself, so F − (0) is a submanifold of U by the implicit function theorem.By construction it contains both the Ricci-flat metrics in U and Z D k +1 ∩ U . Now Z × D k +1 → F − (0) (11)is an open map near ( g, id ) by the submersion theorem (it is smooth because elements of Z are). This implies that any smooth Ricci-flat metric g ′ near g is D k +1 -equivalent to an elementof Z , which must in fact lie in Q because Ricci-flatness is a diffeomorphism-invariant property.Since isometries between smooth Riemannian metrics are smooth (see Myers and Steenrod [15,Theorem 8]), g ′ is in fact D -equivalent to an element of Q . In other words, Q → W is open.Proposition 3.9 implies that in fact Q → W is injective up to the action of the stabiliser I g and, since I g is compact, that the action on Q factors through a finite group (cf. [2, 12 . (cid:3) Remark . Clearly the argument would give the same result even if we were to consider themoduli space of Ricci-flat metrics given by dividing by the action of the full diffeomorphism groupof M . Remark . In [10, Lemma 2 .
6] Koiso uses instead of S a slice constructed by Ebin [6], andshows that any Einstein metric in this slice is smooth.3.4. Proof of theorem I.
Let G be one of the Ricci-flat holonomy groups, M a compact G -man-ifold, Γ(Λ G T ∗ M ) the space of G -structures on M and m : Γ(Λ G T ∗ M ) → Γ( S T ∗ M ) , χ g χ (12)the natural map that sends a G -structure to the metric it defines. In order to prove theorem I weshow first that for any torsion-free G -structure χ the derivative of m maps the tangent space tothe pre-moduli space R at χ onto the space ε ( g χ ) of infinitesimal Ricci-flat deformations.The tangent space to Γ(Λ G T ∗ M ) at χ is the space of differential forms Γ( E χ ), where E χ ⊆ Λ ∗ T ∗ M is a vector subbundle associated to the G -structure defined by χ . Fibre-wise Λ G T ∗ M is a GL ( R n )-orbit and E χ is the tangent space gl n χ to the orbit. Because m is GL ( R n )-equivariantits derivative takes aχ ag χ for any a ∈ gl n , which maps onto the fibre of S T ∗ M . Hence thederivative Dm χ : Γ( E χ ) → Γ( S T ∗ M ) (13)is surjective. Furthermore, the derivative is G -equivariant with respect to the G -structure definedby χ . Since △ L is the Lichnerowicz Laplacian on S T ∗ M , lemma 2.6 implies that the diagrambelow commutes. Γ( E χ ) Dm χ ✲ Γ( S T ∗ M )Γ( E χ ) △ ❄ Dm χ ✲ Γ( S T ∗ M ) △ L ❄ Hence
Lemma 3.13. If χ is a torsion-free G -structure then Dm χ maps the harmonic sections of E χ onto the space ε ( g χ ) of infinitesimal Ricci-flat deformations. So let χ be any torsion-free G -structure on M and R the pre-moduli space of torsion-free G -structures near χ . As described in subsection 3.3, there is a slice at g χ for the D -action on themetrics, the Ricci-flat metrics in the slice are a real analytic subset of a submanifold Z , and thetangent space to Z at χ is ε ( g χ ). Let P : F − (0) → Z be the composition of a smooth local rightinverse to the submersion (11) with the projection to the first factor. F − (0) contains the Ricci-flatmetrics near g χ , and P can be viewed as a local projection to the slice: P ( g ′ ) is D -equivalent to g ′ for any Ricci-flat g ′ close to g χ . Then P ◦ m : R → Z (14)is a well-defined smooth map and lemma 3.13 means that its derivative at χ is surjective. Thereforeevery element of ε ( g χ ) is tangent to a path of Ricci-flat metrics, so Q is a manifold. By thesubmersion theorem, W G (the image of M G in W ) contains a neighbourhood of g D .The pre-images of g χ under m are defined by differential forms which are harmonic with respectto g χ . By Hodge theory they represent distinct cohomology classes. Let I g χ ⊆ D be the isometriesof g χ isotopic to the identity. Because I g χ acts trivially on cohomology it must fix the fibre over of m over g χ , so I g χ = I χ . Now, if g ′ ∈ Q then g ′ = φ ∗ m ( χ ′ ) for some χ ′ ∈ R and φ ∈ D k +1 because(14) is a submersion. As I χ acts trivially on R by proposition 3.2 it follows that the conjugate I φg χ fixes g ′ . But then I φg χ ⊆ I g ′ ⊆ I g χ by proposition 3.9, so in fact I φg χ = I g χ . Hence I g χ fixes any g ′ ∈ Q .Now theorem 3.10 implies that Q is homeomorphic to a neighbourhood of W . Thus W G is amanifold near g D and the proof of theorem I is complete.3.5. The asymptotically cylindrical case.
The proof of theorem I only used the compactnessassumption to access certain deformation results for G -structures and Ricci-flat metrics. For thecases G = G and Spin (7) there are pre-moduli spaces of EAC G -structures, with propertiesanalogous to proposition 3.1.If M is an EAC G -manifold, let M G denote the quotient of the space of torsion-free EAC G -structures on M by the group D of EAC diffeomorphisms of M isotopic to the identity. Proposition 3.14.
Let G = Spin (7) or G , M an EAC G -manifold and χ a torsion-free EAC G -structure on M . Then there is a submanifold R of the space of C G -structures such that (i) the elements of R are smooth EAC torsion-free G -structures, (ii) the tangent space to R at χ is the space of bounded harmonic sections of E χ , (iii) the natural map R → M G is a homeomorphism onto a neighbourhood of χ D in M G .Proof. See [17, §
6] for the G case, and [16, § Spin (7) case. (cid:3)
ICCI-FLAT DEFORMATIONS OF METRICS WITH EXCEPTIONAL HOLONOMY 11
In order to prove the theorem I ′ , the EAC version of theorem I, it therefore suffices to explainhow to set up the deformation theory for EAC Ricci-flat metrics. Below we define the slices withsame equations as in the compact case in § G -manifolds in [17, § M n be a manifold with cylindrical ends and cross-section X n − . Let W be the quotientof the space of EAC Ricci-flat metrics (with any exponential rate) by the group D of EAC dif-feomorphisms of M isotopic to the identity. We pick an EAC Ricci-flat metric g on M and studya neighbourhood of g D in W . By definition, the asymptotic limit of g is a cylindrical metric dt + g X on X × R , where g X is a Ricci-flat metric on X .We work with weighted H¨older spaces of sections. Let k ≥ α ∈ (0 , δ > g . The metric g defines a Hodge Laplacian on 1-forms and a LichnerowiczLaplacian on symmetric bilinear forms, which are both asymptotically translation-invariant oper-ators. We require that δ is small enough that the Laplacians are Fredholm on C k,αδ spaces, as wemay according to theorem 2.8.We proved in § Z ⊂ C k,α ( S T ∗ X ) which containsrepresentatives of all diffeomorphism classes of Ricci-flat metrics on X close to g X . Its tangentspace T g X Z = ε ( g X ) is the space of Lichnerowicz harmonic sections of S T ∗ X .Let M kZ denote the space of C k,α metrics on M which are C k,αδ -asymptotic to cylindrical metrics dt + g X such that g X ∈ Z . If ρ is a cut-off function for the cylinder then ρZ can be identified witha space of bilinear forms on M , and M kZ is an open subset M kZ ⊂ C k,αδ ( S ∗ T ∗ M ) + ρZ. Similarly let D k +1 Z be the set of EAC diffeomorphisms with rate δ which are asymptotic to elementsof the isometry group I g X of g X . Then M kZ contains representatives of all diffeomorphism classes ofRicci-flat metrics near g and, because Z is I g X -invariant, proposition 3.9 implies that any isometrybetween elements of M kZ must lie in D k +1 Z (a similar argument for simplifying the problem by aslice at the boundary was used to study the moduli space of torsion-free EAC G -structures in[17, Lemma 6.24]). We therefore identify a slice in M kZ for the action of D k +1 Z at g . The tangentspace to M kZ at g is T g M kZ = C k,αδ ( S ∗ T ∗ M ) ⊕ ρε ( g X ) . The tangent space at the identity of D k +1 Z corresponds to vector fields which are C k,αδ -asymptoticto translation-invariant Killing vector fields on the cylinder, i.e. to elements of ( H ∞ ) ♯ , where H ∞ denotes the translation-invariant harmonic 1-forms on the cylinder X × R . By proposition 3.4 thetangent space to the D k +1 Z -orbit at g is δ ∗ g ( C k,αδ (Λ ) ⊕ ρ H ∞ ) . Let K be the kernel of 2 δ g + d tr g in T g M kZ . Lemma 3.15.
Let M be a Ricci-flat EAC manifold with a single end. Then T g M kZ = K ⊕ δ ∗ g ( C k,αδ (Λ ) ⊕ ρ H ∞ ) . (15) Proof. (2 δ g + d tr g ) δ ∗ = △ g according to lemma 3.5, so it suffices to show that the image of2 δ g + d tr g : T g M kZ → C k − ,αδ (Λ ) is contained in the image of △ : C k +1 ,αδ (Λ ) ⊕ ρ H ∞ → C k − ,αδ (Λ ) . It follows from theorem 2.8 that this has index 0, so its image is the L -orthogonal complementto its kernel H , the space of bounded harmonic 1-forms.Now, if h ∈ T g M kZ and β ∈ H then the difference between <δ g h, β> and
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Department of Mathematics, Imperial College London, London SW7 2AZ, UK
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