A moment map interpretation of the Ricci form, Kähler--Einstein structures, and Teichmüller spaces
aa r X i v : . [ m a t h . S G ] A p r A moment map interpretation of the Ricci form,K¨ahler–Einstein structures, and Teichm¨uller spaces
Oscar Garc´ıa-Prada and Dietmar Salamon
This paper is dedicated to the memory of Boris Dubrovin.
Abstract.
This paper surveys the role of moment maps in K¨ahler geometry.The first section discusses the Ricci form as a moment map and then moveson to moment map interpretations of the K¨ahler–Einstein condition and thescalar curvature (Quillen–Fujiki–Donaldson). The second section examines theramifications of these results for various Teichm¨uller spaces and their Weil–Petersson symplectic forms and explains how these arise naturally from theconstruction of symplectic quotients. The third section discusses a symplecticform introduced by Donaldson on the space of Fano complex structures.
1. The Ricci form
This section explains how the Ricci form appears as a moment map for theaction of the group of exact volume preserving diffeomorphisms on the space ofalmost complex structures. A direct consequence of this observation is the Quillen–Fujiki–Donaldson Theorem about the scalar curvature as a moment map for theaction of the group of Hamiltonian symplectomorphisms on the space of compatiblealmost complex structures on a symplectic manifold. This section also discusses howthe K¨ahler–Einstein condition can be interpreted as a moment map equation.
Let M be a closed oriented 2 n -manifold equipped with a positive volume form ρ ∈ Ω n ( M ). Then the space J ( M )of all almost complex structures on M that are compatible with the orientationcan be thought of as an infinite-dimensional symplectic manifold. Its tangent spaceat J ∈ J ( M ) is the space of all complex anti-linear endomorphisms b J : T M → T M of the tangent bundle (see [ , Section 2]) and thus can be identified with thespace Ω , J ( M, T M ) of complex anti-linear 1-forms on M with values in the tangentbundle. The symplectic form Ω ρ is given by(1.1) Ω ρ,J ( b J , b J ) := Z M trace (cid:16) b J J b J (cid:17) ρ for J ∈ J ( M ) and b J , b J ∈ T J J ( M ) = Ω , J ( M, T M ). Mathematics Subject Classification.
Key words and phrases. moment map, Ricci form, K¨ahler–Einstein, Teichm¨uller space.
The symplectic form is preserved by the action of the group Diff(
M, ρ ) ofvolume preserving diffeomorphisms. Denote the identity component by Diff ( M, ρ )and the subgroup of exact volume preserving diffeomorphisms (that are isotopic tothe identity via an isotopy that is generated by a smooth family of exact divergence-free vector fields) by Diff ex ( M, ρ ).Consider the submanifold J ( M ) ⊂ J ( M ) of all almost complex structuresthat are compatible with the orientation and have real first Chern class zero. It wasshown in [ ] that the action of Diff ex ( M, ρ ) on J ( M ) is Hamiltonian and thattwice the Ricci form appears as a moment map. To make this precise, note thatthe Lie algebra of Diff ex ( M, ρ ) is the space of exact divergence-free vector fieldsand can be identified with the quotient space Ω n − ( M ) / ker d via the correspon-dence Ω n − ( M ) → Vect ex ( M, ρ ) : α Y α , defined by ι ( Y α ) ρ = dα. The dual space of the quotient Ω n − ( M ) / ker d can formally be thought of asthe space of exact 2-forms, in that every exact 2-form τ ∈ d Ω ( M ) gives rise to acontinuous linear functional Ω n − ( M ) / ker d → R : [ α ] R M τ ∧ α .The Ricci form
Ric ρ,J ∈ Ω ( M ) associated to a volume form ρ and an almostcomplex structure J , both inducing the same orientation of M , is defined byRic ρ,J ( u, v ) := trace (cid:0) JR ∇ ( u, v ) (cid:1) + trace (cid:0) ( ∇ u J ) J ( ∇ v J ) (cid:1) + dλ ∇ J ( u, v )(1.2)for u, v ∈ Vect( M ). Here ∇ is a torsion-free connection on T M that preserves thevolume form ρ and the 1-form λ ∇ J ∈ Ω ( M ) is defined by λ ∇ J ( u ) := trace (cid:0) ( ∇ J ) u (cid:1) for u ∈ Vect( M ). The Ricci form is independent of the choice of the torsion-free ρ -connection used to define it and it is closed and represents the cohomologyclass 2 πc R ( J ). Its dependence on the volume form is governed by the identity(1.3) Ric e f ρ,J = Ric ρ,J + d ( df ◦ J ) . for J ∈ J ( M ) and f ∈ Ω ( M ), and the map ( ρ, J ) Ric ρ,J is equivariant underthe action of the diffeomorphism group, i.e.(1.4) Ric φ ∗ ρ,φ ∗ J = φ ∗ Ric ρ,J for all J ∈ J ( M ) and all φ ∈ Diff( M ).The definition of the Ricci form in (1.2) arises as a special case of a general mo-ment map identity in [ ] for sections of certain SL(2 n , R ) fiber bundles. If ρ is thevolume form of a K¨ahler metric and ∇ is the Levi-Civita connection, then ∇ J = 0and hence the last two terms in (1.2) vanish and Ric ρ,J is the standard Ricci form.In general, the second summand in (1.2) is a correction term which gives rise to aclosed 2-form that represents 2 π times the first Chern class, and the last summandis a further correction term that makes the Ricci form independent of the choice ofthe torsion-free ρ -connection ∇ . If J is compatible with a symplectic form ω and ∇ is the Levi-Civita connection of the Riemannian metric ω ( · , J · ), then λ ∇ J = 0. Inthe integrable case the 2-form i Ric ρ,J is the curvature of the Chern connection onthe canonical bundle associated to the Hermitian structure determined by ρ andhence is a (1 , ICCI FORM AND TEICHM¨ULLER SPACES 3
Theorem ]) . The action of the group
Diff ex ( M, ρ ) on the space J ( M ) with the symplectic form (1.1) is a Hamiltonian group action and is generated bythe Diff(
M, ρ ) -equivariant moment map J ( M ) → d Ω ( M ) : J ρ,J , i.e. (1.5) Z M d Ric ρ ( J, b J ) ∧ α = Ω ρ,J ( b J, L Y α J ) for all J ∈ J ( M ) , all b J ∈ Ω , J ( M, T M ) and all α ∈ Ω n − ( M ) , where d Ric ρ ( J, b J ) := ddt (cid:12)(cid:12) t =0 Ric ρ,J t for any smooth path R → J ( M ) : t J t satisfying J = J and ddt (cid:12)(cid:12) t =0 J t = b J . Theorem 1.1 is based on ideas in [ ]. We emphasize that equation (1.5) doesnot require the vanishing of the first Chern class. Its proof in [ ] relies on theconstruction of a 1-form Λ ρ on J ( M ) with values in the space of 1-forms on M .For J ∈ J ( M ) and b J ∈ Ω , J ( M, T M ) the 1-form Λ ρ ( J, b J ) ∈ Ω ( M ) is defined by(1.6) (cid:0) Λ ρ ( J, b J ) (cid:1) ( u ) := trace (cid:0) ( ∇ b J ) u + b JJ ∇ u J (cid:1) for u ∈ Vect( M ), where ∇ is a torsion-free ρ -connection. As before, Λ ρ ( J, b J ) isindependent of the choice of ∇ . Moreover, Λ ρ satisfies the following identities. Proposition ]) . Let J ∈ J ( M ) , b J ∈ Ω , J ( M, T M ) , and v ∈ Vect( M ) .Denote the divergence of v by f v := dι ( v ) ρ/ρ . Then d (cid:0) Λ ρ ( J, b J ) (cid:1) = 2 d Ric ρ ( J, b J ) , (1.7) Z M Λ ρ ( J, b J ) ∧ ι ( v ) ρ = Z M trace (cid:0) b JJ L v J (cid:1) ρ, (1.8) Λ ρ ( J, L v J ) = 2 ι ( v )Ric ρ,J − df v ◦ J + df Jv . (1.9)For a proof of Proposition 1.2 see [ , Theorems 2.6 & 2.7], and note thatequation (1.5) in Theorem 1.1 follows directly from (1.7) and (1.8) with v = Y α . Remark . Two useful equations (see [ , Lemma 2.12]) are(1.10) L X J = 2 J ¯ ∂ J X, Λ ρ ( J, b J ) = ι (2 J ¯ ∂ ∗ J b J ∗ ) ω for J ∈ J ( M ), b J ∈ Ω , J ( M, T M ), X ∈ Vect( M ). Here ω is a nondegenerate 2-formon M such that ω n / n ! = ρ and h· , ·i := ω ( · , J · ) is a Riemannian metric.For any Hamiltonian group action the zero set of the moment map is invariantunder the group action and its orbit space is called the Marsden–Weinstein quotient.In the case at hand this quotient is the space of exact volume preserving isotopyclasses of Ricci-flat almost complex structures given by W ( M, ρ ) := J ( M, ρ ) / Diff ex ( M, ρ ) , J ( M, ρ ) := { J ∈ J ( M ) | Ric ρ,J = 0 } . (1.11)In the finite-dimensional setting it follows directly from the definitions that anelement of the zero set of the moment map is a regular point for the moment map(i.e. its derivative is surjective) if and only if the isotropy subgroup is discrete. Itwas shown in [ , Theorem 2.11] that this carries over to the present situation. OSCAR GARC´IA-PRADA AND DIETMAR SALAMON
Proposition ]) . Fix an element J ∈ J ( M ) . (i) Let b λ ∈ Ω ( M ) . Then there exists a b J ∈ Ω , J ( M, T M ) such that d Ric ρ ( J, b J ) = d b λ if and only if R M d b λ ∧ α = 0 for all α ∈ Ω n − ( M ) with L Y α J = 0 . (ii) Let b J ∈ Ω , J ( M, T M ) . Then there exists a α ∈ Ω n − ( M ) such that L Y α J = b J if and only if Ω ρ,J ( b J, b J ′ ) = 0 for all b J ′ ∈ Ω , J ( M, T M ) with d Ric ρ ( J, b J ′ ) = 0 . Call an almost complex structure J ∈ J ( M ) regular if there is no nonzeroexact divergence-free J -holomorphic vector field. The set of regular almost complexstructures is open and the next proposition gives a regularity criterion. It showsthat every K¨ahlerable complex structure with real first Chern class zero is regular. Proposition ]) . Assume that J ∈ J ( M ) satisfies Ric ρ,J = 0 and iscompatible with a symplectic form ω ∈ Ω ( M ) such that ω n / n ! = ρ and the ho-momorphism H ( M ; R ) → H n − ( M, R ) : [ λ ] [ λ ∧ ω n − / ( n − is bijective.Then L Y α J = 0 implies Y α = 0 for every α ∈ Ω n − ( M ) . Proof.
Assume L Y α J = 0. Then ι ( Y α ) ω is harmonic by [ , Lemma 3.9(ii)]and is exact because ι ( Y α ) ω ∧ ω n − / ( n − dα . Thus Y α = 0. (cid:3) By part (i) of Proposition 1.4 an almost complex structure J ∈ J ( M ) is reg-ular if and only if the linear map Ω , J ( M, T M ) → d Ω ( M ) : b J d Ric ρ ( J, b J ) is sur-jective. By the implicit function theorem in appropriate Sobolev completions thisimplies that the regular part of J ( M, ρ ) is a co-isotropic submanifold of J ( M )whose isotropic fibers are the Diff ex ( M, ρ )-orbits. One would like to deduce thatthe regular part of W ( M, ρ ) is a symplectic orbifold with the tangent spaces T [ J ] ρ W ( M, ρ ) = n b J ∈ Ω , J ( M, T M ) | d Ric ρ ( J, b J ) = 0 o {L Y α J | α ∈ Ω n − ( M ) } at regular elements J ∈ J ( M, ρ ). Indeed, the 2-form (1.1) is nondegenerate onthis quotient by part (ii) of Proposition 1.4. However, the action of Diff ex ( M, ρ )on J ( M, ρ ) is not always proper and the quotient W ( M, ρ ) need not be Hausdorff.The archetypal example is the K3-surface [
34, 58 ]. Example . Let (
M, J ) be a K3-surface that admits an embedded holo-morphic sphere C with self-intersection number C · C = −
2, and let τ : M → M be the Dehn twist about C . Then there exists a smooth family of complex struc-tures { J t } t ∈ C and a smooth family of diffeomorphisms { φ t ∈ Diff ( M ) } t ∈ C \{ } suchthat J = J and φ ∗ t J t = τ ∗ J − t for t = 0. Thus the complex structures J t and τ ∗ J − t represent the same equivalence class in W ( M ), however, their limits lim t → J t = J and lim t → τ ∗ J − t = τ ∗ J do not represent the same class in W ( M ) because the ho-mology class [ C ] belongs to the effective cone of J while the class − [ C ] belongs tothe effective cone of τ ∗ J . This shows that the action of Diff ( M ) on J ( M ) isnot proper and neither is the action of Diff ( M, ρ ) = Diff ex ( M, ρ ) on J ( M, ρ ). Let M be a closed oriented 2 n -mani-fold and fix a nonzero real number ~ . A tame symplectic Einstein structure on M is a pair ( ω, J ) consisting of a symplectic form ω ∈ Ω ( M ) and an almostcomplex structure J tamed by ω (i.e. ω ( v, Jv ) > = v ∈ T M ) such that(1.12) Ric ρ,J = ω/ ~ , ρ := ω n / n ! . ICCI FORM AND TEICHM¨ULLER SPACES 5
Every such structure satisfies 2 π ~ c R ( ω ) = [ ω ]. In the case dim( M ) = 4 the integralfirst Chern class of a symplectic form ω depends only on the cohomology class of ω (see [ , Proposition 13.3.11]), while in higher dimensions this is an open question.The purpose of this subsection is to exhibit the space of equivalence classes oftame symplectic Einstein structures in a fixed cohomology class [ ω ] = a ∈ H ( M ; R )and with a fixed volume form ω n / n ! = ρ , modulo the action of the group of exactvolume preserving diffeomorphisms, as a symplectic quotient.A pair ( a, ρ ) consisting of a cohomology class a ∈ H ( M ; R ) and a positivevolume form ρ is called a Lefschetz pair if it satisfies the following conditions. (V) V := h a n / n ! , [ M ] i > and R M ρ = V . (L) The homomorphism H ( M ; R ) → H n − ( M ; R ) : b a n − ∪ b is bijective. Fix a Lefschetz pair ( a, ρ ). Then the space S a,ρ := (cid:8) ω ∈ Ω ( M ) (cid:12)(cid:12) dω = 0 , [ ω ] = a, ω n / n ! = ρ (cid:9) of symplectic forms on M in the cohomology class a with volume form ρ is aninfinite-dimensional manifold whose tangent space at ω ∈ S a,ρ is given by T ω S a,ρ = (cid:8)b ω ∈ d Ω ( M ) (cid:12)(cid:12) b ω ∧ ω n − = 0 (cid:9) . The proof uses the fact that the map S a → V a : ω ω n / n ! from the space S a ofsymplectic forms in the class a to the space V a of volume forms with total volume V is a submersion. (It is surjective by Moser isotopy whenever S a = ∅ .) It was shownby Trautwein [ , Lemma 6.4.2] that S a,ρ carries a symplectic structure(1.13) Ω ω ( b ω , b ω ) := Z M b λ ∧ b λ ∧ ω n − ( n − ω ∈ S a,ρ and b ω , b ω ∈ T ω S a,ρ , where b λ i ∈ Ω ( M ) is chosen such that(1.14) d b λ i = b ω i , b λ i ∧ ω n − ∈ d Ω n − ( M ) . Here the existence of b λ i and the nondegeneracy of (1.13) both require the Lefschetzcondition (L). In the rational case a result of Fine [ ] shows that S a,ρ is a sym-plectic quotient via the action of the group of gauge transformations on the spaceof connections with symplectic curvature on a suitable line bundle.Now consider the space P ( M, a, ρ ) := (cid:8) ( ω, J ) ∈ S a,ρ × J ( M ) (cid:12)(cid:12) ω ( v, Jv ) > = v ∈ T M (cid:9) of all pairs ( ω, J ) consisting of a symplectic form ω in the class a with volume form ρ and an ω -tame almost complex structure J . This is an open subset of the productspace S a,ρ × J ( M ), and the symplectic forms (1.13) on S a,ρ and (1.1) on J ( M )together determine a natural product symplectic structure on P ( M, a, ρ ), given by(1.15) Ω ω,J (cid:0) ( b ω , b J ) , ( b ω , b J ) (cid:1) := Z M trace (cid:0) b J J b J (cid:1) ω n n ! − ~ Z M b λ ∧ b λ ∧ ω n − ( n − ω, J ) ∈ P ( M, a, ρ ) and ( b ω i , b J i ) ∈ T ( ω,J ) P ( M, a, ρ ) = T ω S a,ρ × Ω , J ( M, T M ),where the b λ i are as in (1.14).Throughout we will use the notation Y α for the exact divergence-free vector fieldassociated to a (2 n − α ∈ Ω n − ( M ) via ι ( Y α ) ρ = dα . When the choice ofthe symplectic form ω is clear from the context, we will use the notation v H for theHamiltonian vector field associated to a function H ∈ Ω ( M ) via ι ( v H ) ω = dH . OSCAR GARC´IA-PRADA AND DIETMAR SALAMON
Theorem
Trautwein) . The symplectic form (1.15) on P ( M, a, ρ ) ispreserved by the action of Diff(
M, ρ ) . The action of the subgroup Diff ex ( M, ρ ) is aHamiltonian group action and is generated by the Diff(
M, ρ ) -equivariant momentmap P ( M, a, ρ ) → d Ω ( M ) : ( ω, J ) (cid:0) Ric ρ,J − ω/ ~ (cid:1) , i.e. (1.16) Z M (cid:0)d Ric ρ ( J, b J ) − b ω/ ~ (cid:1) ∧ α = Ω ω,J (cid:0) ( b ω, b J ) , ( L Y α ω, L Y α J ) (cid:1) for all ( ω, J ) ∈ P ( M, a, ρ ) , all ( b ω, b J ) ∈ T ( ω,J ) P ( M, a, ρ ) , and all α ∈ Ω n − ( M ) . Proof.
Equation (1.16) follows from Theorem 1.1 and the identity Z M b ω ∧ α = Z M b λ ∧ ι ( Y α ) ω ∧ ω n − ( n − ω ( b ω, L Y α ω )for all ω ∈ S a,ρ , all b ω = d b λ ∈ T ω S a,ρ , and all α ∈ Ω n − ( M ). (cid:3) Proposition . The action of
Diff ex ( M, ρ ) on P ( M, a, ρ ) is proper. Proof.
The properness proof by Fujiki–Schumacher [ ] carries over to thepresent situation. Choose sequences ( ω i , J i ) ∈ P ( M, a, ρ ) and φ i ∈ Diff ex ( M, ρ )such that the limits ( ω, J ) = lim t →∞ ( ω i , J i ) and ( ω ′ , J ′ ) = lim t →∞ ( φ ∗ i ω i , φ ∗ i J i ) ex-ist in the C ∞ topology and both belong to P ( M, a, ρ ). Define the Riemannian met-rics g i by g i ( u, v ) := ( ω i ( u, J i v ) + ω i ( v, J i u )) and similarly for g, g ′ . Then g i con-verges to g and φ ∗ i g i converges to g ′ in the C ∞ topology. Thus by [ , Lemma 3.8] asubsequence of φ i converges to a diffeomorphism φ ∈ Diff(
M, ρ ) in the C ∞ topology.Now the flux homomorphism Flux ρ : π (Diff ( M, ρ )) → H n − ( M ; R ) has a discreteimage by [ , Exercise 10.2.23(v)]. Thus it follows from standard arguments asin [ , Theorem 10.2.5 & Proposition 10.2.16] that Diff ex ( M, ρ ) is a closed sub-group of Diff(
M, ρ ) with respect to the C ∞ topology. Hence φ ∈ Diff ex ( M, ρ ). (cid:3) In the present setting the Marsden–Weinstein quotient is the space of exactvolume preserving isotopy classes of tame symplectic Einstein structures given by W SE ( M, a, ρ ) := P SE ( M, a, ρ ) / Diff ex ( M, ρ ) , P SE ( M, a, ρ ) := { ( ω, J ) ∈ P ( M, a, ρ ) | Ric ρ,J = ω/ ~ } . (1.17)The next result is the analogue of Proposition 1.4 in the symplectic Einstein setting.In particular, part (i) asserts that a pair ( ω, J ) ∈ P ( M, a, ρ ) is a regular point forthe moment map if and only if the isotropy subgroup is discrete. While the finite-dimensional analogue follows directly from the definitions, in the present situationthe proof requires elliptic regularity (in the guise of Proposition 1.4).
Proposition . Fix a pair ( ω, J ) ∈ P ( M, a, ρ ) . (i) Let b λ ∈ Ω ( M ) . Then there exists a pair ( b ω, b J ) ∈ T ( ω,J ) P ( M, a, ρ ) that sat-isfies d Ric ρ ( J, b J ) − b ω/ ~ = d b λ if and only if R M b λ ∧ ι ( v H ) ρ = 0 for all H ∈ Ω ( M ) with L v H J = 0 . (ii) Let ( b ω, b J ) ∈ T ( ω,J ) P ( M, a, ρ ) . Then there exists a (2 n − -form α ∈ Ω n − ( M ) that satisfies L Y α ω = b ω and L Y α J = b J if and only if Ω ω,J (cid:0) ( b ω, b J ) , ( b ω ′ , b J ′ ) (cid:1) = 0 forall ( b ω ′ , b J ′ ) ∈ T ( ω,J ) P ( M, a, ρ ) with d Ric ρ ( J, b J ′ ) = b ω ′ / ~ . The necessity of the conditions in (i) and (ii) follows directly from (1.16). Theproof of the converse implications relies on the following three lemmas, which allowus to reduce the result to Proposition 1.4.
ICCI FORM AND TEICHM¨ULLER SPACES 7
Lemma . Let ( ω, J ) ∈ P ( M, a, ρ ) and let α ∈ Ω n − ( M ) . Then Y α isHamiltonian if and only if R M λ ∧ dα = 0 for every -form λ with dλ ∧ ω n − = 0 . Proof.
That the condition is necessary follows directly from the definitions.Conversely, assume that R M λ ∧ dα = 0 for all λ ∈ Ω ( M ) with dλ ∧ ω n − = 0. Let β be a closed (2 n − λ ∈ Ω ( M ) with β = λ ∧ ω n − / ( n − Z M β ∧ (( ∗ dα ) ◦ J ) = Z M λ ∧ (( ∗ dα ) ◦ J ) ∧ ω n − ( n − Z M λ ∧ dα = 0 . Hence ( ∗ dα ) ◦ J is an exact 1-form. Choose H ∈ Ω ( M ) such that ( ∗ dα ) ◦ J = dH .Then ∗ dα = − dH ◦ J , hence dα = ∗ ( dH ◦ J ) = dH ∧ ω n − / ( n − ι ( v H ) ρ andso Y α = v H is a Hamiltonian vector field. (cid:3) Lemma . Let ( ω, J ) ∈ P ( M, a, ρ ) and let Y ⊂ Vect ex ( M, ρ ) be a finite-di-mensional subspace that contains no nonzero Hamiltonian vector field. Then, forevery linear functional Φ : Y → R , there exists a -form λ such that dλ ∧ ω n − = 0 and R M λ ∧ ι ( Y ) ρ = Φ( Y ) for all Y ∈ Y . Proof.
Define L := { λ ∈ Ω ( M ) | dλ ∧ ω n − = 0 } and consider the linearmap L → Y ∗ : λ Φ λ defined by Φ λ ( Y ) := R M λ ∧ ι ( Y ) ρ for λ ∈ L and Y ∈ Y .Then the dual map Y → L ∗ is injective by assumption and Lemma 1.10. Since Y is finite-dimensional, this implies that the map L → Y ∗ is surjective. (cid:3) Lemma . Let ( ω, J ) ∈ P ( M, a, ρ ) and let α ∈ Ω n − ( M ) . Then the follow-ing assertions are equivalent. (a) There exists a function H ∈ Ω ( M ) such that L v H J = L Y α J . (b) If λ ∈ Ω ( M ) satisfies dλ ∧ ω n − = 0 and R M dλ ∧ β = 0 for all β ∈ Ω n − ( M ) with L Y β J = 0 , then R M dλ ∧ α = 0 . Proof.
Assume (a) and define β := α − Hω n − / ( n − Y β = Y α − v H ,hence L Y β J = 0 by (a), and so each λ as in (b) satisfies Z M dλ ∧ α = Z M dλ ∧ (cid:18) β + H ω n − ( n − (cid:19) = 0 . Conversely, assume (a) does not hold and choose a subspace Y ⊂ Vect ex ( M, ρ )such that { Y β | L Y β J = 0 } = Y ⊕ { v H | L v H J = 0 } . Then Y := Y ⊕ R Y α doesnot contain nonzero Hamiltonian vector fields and so, by Lemma 1.11 there isa λ ∈ Ω ( M ) such that dλ ∧ ω n − = 0, R M λ ∧ ι ( Y α ) ρ = 1, and R M λ ∧ ι ( Y ) ρ = 0for all Y ∈ Y . This implies R M dλ ∧ β = 0 for all β ∈ Ω n − ( M ) with L Y β J = 0and R M dλ ∧ α = 1. Hence (b) does not hold. (cid:3) Proof of Proposition 1.9.
We prove part (i). Assume that b λ ∈ Ω ( M ) sat-isfies R M b λ ∧ ι ( v H ) ρ = 0 for all H ∈ Ω ( M ) with L v H J = 0. Then by Lemma 1.11there exists a λ ∈ Ω ( M ) such that dλ ∧ ω n − = 0 and R M ( λ + b λ ) ∧ ι ( Y α ) ρ = 0for all α ∈ Ω n − ( M ) with L Y α J = 0. By part (i) of Proposition 1.4 there existsa b J ∈ Ω , J ( M, T M ) such that d Ric ρ ( J, b J ) = d ( λ + b λ ). This proves (i) with b ω := ~ dλ .To prove (ii), let ( b ω, b J ) ∈ T ( ω,J ) P ( M, a, ρ ) such that Ω ω,J (cid:0) ( b ω, b J ) , ( b ω ′ , b J ′ ) (cid:1) = 0for all ( b ω ′ , b J ′ ) ∈ T ( ω,J ) P ( M, a, ρ ) with d Ric ρ ( J, b J ′ ) = b ω ′ / ~ . Then by part (ii) ofProposition 1.4 there exist α, β with b ω = dι ( Y α ) ω and b J = L Y β J . Let λ ∈ Ω ( M ) OSCAR GARC´IA-PRADA AND DIETMAR SALAMON such that dλ ∧ ω n − = 0 and R M dλ ∧ α ′ = 0 for all α ′ ∈ Ω n − ( M ) with L Y α ′ J = 0.By part (i) of Proposition 1.4 choose b J ′ such that d Ric ρ ( J, b J ′ ) = dλ . Then2 Z M dλ ∧ ( β − α ) = Z M trace (cid:0) b J ′ J L Y β J (cid:1) ω n n ! − ~ Z M ~ λ ∧ (cid:0) ι ( Y α ) ω (cid:1) ∧ ω n − ( n − ω,J (cid:0) ( b ω ′ , b J ′ ) , ( b ω, b J ) (cid:1) = 0 , where b ω ′ := ~ dλ = ~ d Ric ρ ( J, b J ′ ). Thus by Lemma 1.12 there exists a function H suchthat L Y β − α J = L v H J , so L Y α + v H J = b J and dι ( Y α + v H ) ω = b ω . This proves (ii). (cid:3) Call a pair ( ω, J ) ∈ P ( M, a, ρ ) regular if there are no nonzero J -holomorphicHamiltonian vector fields. By part (i) of Proposition 1.9 a pair ( ω, J ) is regularif and only if the map T ( ω,J ) P ( M, a, ρ ) → d Ω ( M ) : ( b ω, b J ) d Ric ρ ( J, b J ) − b ω/ ~ is surjective. Thus by Proposition 1.8 (and a suitable local slice theorem thatrequires a Nash–Moser type proof) the regular part of W SE ( M, a, ρ ) is a sym-plectic orbifold whose tangent space at the equivalence class of a regular ele-ment ( ω, J ) ∈ P SE ( M, a, ρ ) is the quotient T [ ω,J ] ρ W SE ( M, a, ρ ) = (cid:8) ( b ω, b J ) (cid:12)(cid:12) b ω ∧ ω n − = 0 , d Ric ω ( J, b J ) = b ω/ ~ (cid:9) { ( L Y α ω, L Y α J ) | α ∈ Ω n − ( M ) } . The 2-form (1.15) is nondegenerate on this quotient by part (ii) of Proposition 1.9.
Remark
Compatible pairs) . The space C ( M, a, ρ ) ⊂ P ( M, a, ρ ) ofcompatible pairs is a submanifold of P ( M, a, ρ ) and ( b ω, b J ) ∈ T ( ω,J ) P ( M, a, ρ ) istangent to C ( M, a, ρ ) at a compatible pair ( ω, J ) if and only if(1.18) b ω ( u, v ) − b ω ( Ju, Jv ) = ω ( b Ju, Jv ) + ω ( Ju, b Jv ) . It is an open question whether the restriction of the 2-form (1.15) to C ( M, a, ρ ) isnondegenerate. A pair ( b ω, b J ) ∈ T ( ω,J ) C ( M, a, ρ ) belongs to its kernel if and only if(1.19) b J + b J ∗ = 0 , d Ric ρ ( J, b J ) = b ω/ ~ . If this holds, then there exists a vector field X ∈ Vect( M ) such that b ω = dι ( X ) ω and b J = ( L X J − ( L X J ) ∗ ) (respectively b J = L X J in the K¨ahler–Einstein case).Thus the solutions of (1.19) form the kernel of a Fredholm operator, so nondegen-eracy is an open condition. It follows also that the restriction of the 2-form (1.15)to C ( M, a, ρ ) is nondegenerate at a K¨ahler–Einstein pair ( ω, J ) if and only if(1.20) L X J + ( L X J ) ∗ = 0 = ⇒ L X J = 0 . Now fix a K¨ahler manifold (
M, ω, J ). Then Proposition 1.2 yields the equation h ( L X J ) ∗ , L X J i = − Z M trace (cid:0) ( L X J ) J ( L JX J ) (cid:1) ρ = − Z M Λ ρ ( J, L X J ) ∧ ι ( JX ) ρ = Z M (cid:16) df X ◦ J − df JX − ι ( X )Ric ρ,J (cid:17) ∧ ι ( JX ) ρ for every vector field X , and this implies the Weitzenb¨ock formula(1.21) kL X J + ( L X J ) ∗ k = kL X J k + Z M (cid:0) f X + f JX − ρ,J ( X, JX ) (cid:1) ρ. In the K¨ahler–Einstein case Ric ρ,J = ω/ ~ with ~ < ~ > ICCI FORM AND TEICHM¨ULLER SPACES 9
Let (
M, ω ) be a closed symplectic 2 n -manifold withthe volume form ρ := ω n / n !. For F, G ∈ Ω ( M ) we denote by v F the Hamiltonianvector field of F and by { F, G } := ω ( v F , v G ) the Poisson bracket. Let J ( M, ω )be the space of all almost complex structures that are compatible with ω , i.e. thebilinear form h· , ·i := ω ( · , J · ) is a Riemannian metric. This is an infinite-dimensionalmanifold whose tangent space at J ∈ J ( M, ω ) is given by T J J ( M, ω ) = n b J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) ω ( J · , b J · ) + ω ( b J · , J · ) = 0 o . Here the condition ω ( J · , b J · ) + ω ( b J · , J · ) = 0 holds if and only if b J is symmetric withrespect to the Riemannian metric h· , ·i = ω ( · , J · ). In particular, L v J is symmetricfor every symplectic vector field v . The symplectic form on J ( M, ω ) is given by(1.22) Ω ω,J ( b J , b J ) := Z M trace (cid:0) b J J b J (cid:1) ω n n !for J ∈ J ( M, ω ) and b J i ∈ T J J ( M, ω ) and the complex structure is b J
7→ − J b J .With these structures J ( M, ω ) is an infinite-dimensional K¨ahler manifold.The group Symp(
M, ω ) of symplectomorphisms acts on J ( M, ω ) by K¨ahlerisometries. By a theorem of Donaldson [ ], the action of the subgroup Ham( M, ω )of Hamiltonian symplectomorphisms on J ( M, ω ) is a Hamiltonian group actionand the scalar curvature appears as a moment map. Earlier versions of this resultwere proved by Quillen (for Riemann surfaces) and Fujiki [ ] (in the integrablecase). Below we derive it as a direct consequence of Theorem 1.1.To explain this, we first observe that the Lie algebra of the group Ham( M, ω ) isthe space of Hamiltonian vector fields and hence can be identified with the quotientspace Ω ( M ) / R . Its dual space can formally be thought of as the space Ω ρ ( M )of all functions f ∈ Ω ( M ) with mean value zero, in that every such functiondetermines a continuous linear functional Ω ( M ) / R → R : [ H ] R M f Hρ . Nowthe scalar curvature of an almost complex structure J ∈ J ( M, ω ) is defined by(1.23) S ω,J ρ := 2Ric ρ,J ∧ ω n − / ( n − . Its mean value is the topological invariant c ω := πV h c ( ω ) ∪ [ ω ] n − ( n − , [ M ] i , V := R M ρ .In the integrable case S ω,J is the standard scalar curvature of the K¨ahler metric. Theorem
Quillen–Fujiki–Donaldson) . The action of
Ham(
M, ω ) onthe space J ( M, ω ) is Hamiltonian and is generated by the Symp(
M, ω ) -equivariantmoment map J ( M, ω ) → Ω ρ ( M ) : J S ω,J − c ω , i.e. (1.24) Z M b S ω ( J, b J ) H ω n n ! = Ω ω,J ( b J, L v H J ) for all J ∈ J ( M, ω ) , all b J ∈ T J J ( M, ω ) , and all H ∈ Ω ( M ) , where b S ω ( J, b J ) := ddt (cid:12)(cid:12) t =0 S ω,J t for any smooth path R → J ( M, ω ) : t J t satisfying J = J and ddt (cid:12)(cid:12) t =0 J t = b J . Proof.
By (1.23) we have Z M b S ω ( J, b J ) H ω n n ! = Z M d Ric ρ ( J, b J ) ∧ H ω n − ( n − Z M trace (cid:0) b JJ L v H J (cid:1) ω n n ! . The last equation follows from Theorem 1.1 with Y α = v H and α = H ω n − ( n − . (cid:3) In the present situation one proves exactly as in Proposition 1.8 that the actionof the group Ham(
M, ω ) on J ( M, ω ) is proper. Here the argument uses a theoremof Ono [ ] which asserts that Ham( M, ω ) is a closed subgroup of Symp(
M, ω )with respect to the C ∞ topology. Now the Marsden–Weinstein quotient is thespace of Hamiltonian isotopy classes of ω -compatible almost complex structureswith constant scalar curvature given by W csc ( M, ω ) := J csc ( M, ω ) / Ham(
M, ω ) , J csc ( M, ω ) := { J ∈ J ( M, ω ) | S ω,J = c ω } . (1.25)The next result is the analogue of Proposition 1.4 in the present setting. Proposition . Fix an element J ∈ J ( M, ω ) . (i) Let f ∈ Ω ( M ) . Then there exists a b J ∈ T J J ( M, ω ) such that b S ω ( J, b J ) = f ifand only if R M f Hρ = 0 for all H ∈ Ω ( M ) with L v H J = 0 . (ii) Let b J ∈ T J J ( M, ω ) . Then there exists an H ∈ Ω ( M ) such that L v H J = b J ifand only if Ω ω,J ( b J, b J ′ ) = 0 for all b J ′ ∈ T J J ( M, ω ) with b S ω ( J, b J ′ ) = 0 . Proof.
That the conditions in (i) and (ii) are necessary follows from (1.24).To prove the converse, define the operator L : Ω ( M ) → Ω ( M ) by L F := b S ω ( J, J L v F J ) = − d ∗ (cid:0) Λ ρ ( J, J L v F J ) ◦ J (cid:1) = d ∗ dd ∗ dF − d ∗ (cid:0) ι ( Jv F )Ric ρ,J ◦ J (cid:1) + d ∗ (cid:0) Λ ρ (cid:0) J, N J ( v F , · ) (cid:1) ◦ J (cid:1) (1.26)for F ∈ Ω ( M ). Here the last equality follows by a calculation which uses the iden-tities f Jv F = − d ∗ dF and N J ( u, v ) = J ( L v J ) u − ( L Jv J ) u for the Nijenhuis tensor.The operator L is a fourth order self-adjoint Fredholm operator and, by (1.24), Z M ( L F ) Gρ = Z M trace (cid:0) ( L v F J )( L v G J ) (cid:1) ρ (1.27)for all F, G ∈ Ω ( M ). Thus H ∈ ker L if and only if L v H J = 0. Hence, if f ∈ Ω ( M )satisfies R M f Hρ = 0 for all H with L v H J = 0, then f ∈ im L , and this proves (i).To prove part (ii), assume that b J ∈ T J J ( M, ω ) satisfies Ω ω,J ( b J , b J ′ ) = 0 forall b J ′ ∈ T J J ( M, ω ) with b S ω ( J, b J ′ ) = 0. Then, by part (ii) of Proposition 1.4,there exists an α such that L Y α J = b J . Now let λ ∈ Ω ( M ) such that dλ ∧ ω n − = 0and R M dλ ∧ β = 0 for all β ∈ Ω n − ( M ) with L Y β J = 0. Choose b J ′ ∈ Ω , J ( M, T M )with d Ric ρ ( J, b J ′ ) = dλ by part (i) of Proposition 1.4. Then b S ω ( J, b J ′ ) = 0 and hence2 Z M dλ ∧ α = Z M d Ric ρ ( J, b J ′ ) ∧ α = Ω ρ,J ( b J ′ , L Y α J ) = Ω ω,J ( b J ′ , b J ) = 0 . Thus by Lemma 1.12 there exists an H ∈ Ω ( M ) such that L v H J = L Y α J = b J . (cid:3) Proposition . Let J ∈ J ( M, ω ) and let b J ∈ T J J ( M, ω ) . Then thereexists a function H ∈ Ω ( M ) such that b S ω ( J, b J − J L v H J ) = 0 . Moreover, L v H J isuniquely determined by this condition. Proof.
Define f ∈ Ω ( M ) by f ρ := d Λ ρ ( J, b J ) ∧ ω n − / ( n − L be asin Proposition 1.15. Then ( L H ) ρ = d Λ ρ ( J, J L v H J ) ∧ ω n − / ( n − Z M f Hρ = Z M d Λ ρ ( J, b J ) ∧ H ω n − ( n − Z M trace (cid:0) b JJ L v H J (cid:1) ρ = 0for all H ∈ ker L . Thus f belongs to the image of L . (cid:3) ICCI FORM AND TEICHM¨ULLER SPACES 11
Call an almost complex structure J ∈ J ( M, ω ) regular if there are no nonzero J -holomorphic Hamiltonian vector fields. By part (i) of Proposition 1.15 J is regularif and only if the map T J J ( M, ω ) → Ω ρ ( M ) : b J b S ω ( J, b J ) is surjective. Hence,since the action is proper, it follows again from a suitable local slice theorem thatthe regular part of the quotient W csc ( M, ω ) is a K¨ahler orbifold. It is infinite-dimensional when dim( M ) >
2, and its tangent space at the equivalence class of aregular element J ∈ J csc ( M, ω ) is the quotient T [ J ] ω W csc ( M, ω ) = n b J ∈ Ω , J ( M, T M ) | b J = b J ∗ , b S ω ( J, b J ) = 0 o {L v H J | H ∈ Ω ( M ) } . (1.28)The 2-form (1.22) is nondegenerate on this quotient by part (ii) of Proposition 1.15and the complex structure is given by [ b J ] ω [ − J ( b J − L v H J )] ω , where H is chosenas in Proposition 1.16 so that b S ω ( J, J b J − J L v H J ) = 0. Corollary . Let J ∈ J ( M, ω ) and F, G ∈ Ω ( M ) . Then (1.29) Ω ρ,J ( L v F J, L v G J ) = Z M S ω,J { F, G } ω n n ! . In particular, if S ω,J = c ω , then the L -norm of the endomorphism L v F J + J L v G J is given by kL v F J + J L v G J k = kL v F J k + kL v G J k . Proof.
Let φ t be the flow of v F . Then S ω,φ ∗ t J = S ω,J ◦ φ t for all t and hencedifferentiation with respect to t yields the identity(1.30) b S ω ( J, L v F J ) = { S ω,J , F } . Insert equation (1.30) into (1.24) with b J = L v F J and H = G to obtain (1.29). (cid:3) Let (
M, ω, J ) be a closed K¨ahler manifold with constant scalar curvature suchthat H ( M ; R ) = 0. Then every holomorphic vector field is the sum of a Hamilton-ian and a gradient vector field by [ , Lemma 3.7(ii)], and for all F, G ∈ Ω ( M ) wehave kL v F + Jv G J k = kL v F J k + kL v G J k by Corollary 1.17. Hence the Lie alge-bra of holomorphic vector fields is the complexification of the Lie algebra of Killingfields and is therefore reductive. This is the content of Matsushima’s Theorem . Remark . Non-integrable almost complex structures in J ( M, ω ) withvanishing scalar curvature need not be Ricci-flat. To see this, assume dim( M ) ≥ H ( M ; R ) = 0, and that there exists a J ∈ J int ( M, ω ) such that Ric ρ,J = 0with ρ := ω n / n !. Then there exists a b J ∈ Ω , J ( M, T M ) such that(1.31) b J = b J ∗ , b S ω ( J, b J ) = 0 , d Ric ρ ( J, b J ) = 0 . Namely, choose a nonzero 1-form b λ such that d ∗ b λ = 0 and d ∗ ( b λ ◦ J ) = 0. Then b λ isnot closed and there is no nonzero J -holomorphic vector field by [ , Lemma 3.9].Thus, by (1.21) with Ric ρ,J = 0 the operator X
7→ L X J + ( L X J ) ∗ is injective andhence, by the closed image theorem the dual operator b J = b J ∗ ¯ ∂ ∗ J b J is surjective.Thus, by Remark 1.3 there exists a b J = b J ∗ ∈ Ω , J ( M, T M ) such that Λ ρ ( J, b J ) = b λ .This implies b S ω ( J, b J ) = − d ∗ ( b λ ◦ J ) and 2 d Ric ρ ( J, b J ) = d b λ , so b J satisfies (1.31).By (1.31) there exists a smooth curve R → J ( M, ω ) : t J t such that J = J and ∂ t | t =0 J t = b J and S ω,J t = 0 for all t . Since d Ric ρ ( J, b J ) = 0, this curve alsosatisfies Ric ρ,J t = 0 for small nonzero t . Note that d Ric ρ ( J, b J ) is not a (1 , ∂ J b J = 0 by [ , Lemma 3.6], and so J t is not integrable for small nonzero t .
2. Teichm¨uller spaces
In this section we consider integrable complex structures and examine the Teich-m¨uller spaces of Calabi–Yau structures, of K¨ahler–Einstein structures, and of con-stant scalar curvature K¨ahler metrics. Such Teichm¨uller spaces have been studiedby many authors, see e.g. [
7, 12, 28, 29, 30, 34, 38, 39, 40, 45, 46, 49, 50,53, 55, 56, 60 ] and the references therein. The regular part of each Teichm¨ullerspace is a finite-dimensional symplectic submanifold of the relevant symplectic quo-tient in Section 1. It thus acquires a natural symplectic structure that descendsto the Weil–Petersson form on the corresponding moduli space (the quotient ofTeichm¨uller space by the mapping class group).In our formulation the ambient manifold M is fixed and the Weil–Peterssonform arises via symplectic reduction of a Hamiltonian group action by an infinite-dimensional group on an infinite-dimensional space with a finite-dimensional quo-tient. In the original algebro-geometric approach the moduli space is directlycharacterized in finite-dimensional terms via a Torelli type theorem and the Weil–Petersson form arises from the natural homogeneous symplectic form on the relevantperiod domain.It seems to be an open question whether there exist closed K¨ahler manifoldsthat admit holomorphic diffeomorphisms that are smoothly isotopic to the identity,but not through holomorphic diffeomorphisms. (For nonK¨ahler examples see [ ].)If they do exist, then the regular parts of the Teichm¨uller spaces examined here areorbifolds rather than manifolds. When discussing Teichm¨uller spaces as manifolds,we tacitly assume that such automorphisms do not exist, as is the case for Riemannsurfaces. Let M be aclosed connected oriented 2 n -manifold. Then the Teichm¨uller space of Calabi–Yau structures on M is the space of isotopy classes of complex structures withreal first Chern class zero and nonempty K¨ahler cone. It is denoted by T ( M ) := J int , ( M ) / Diff ( M ) , J int , ( M ) := (cid:8) J ∈ J int ( M ) (cid:12)(cid:12) c R ( J ) = 0 and J admits a K¨ahler form (cid:9) . (2.1)Associated to a complex structure J ∈ J int , ( M ) there is the Dolbeault complex(2.2) Ω ( M, T M ) ¯ ∂ J −→ Ω , J ( M, T M ) ¯ ∂ J −→ Ω , J ( M, T M ) , where the first operator corresponds to the infinitesimal action of the vector fieldson T J J ( M ) = Ω , J ( M, T M ) by Remark 1.3, and the second operator correspondsto the derivative of the map which assigns to an almost complex structure J itsNijenhuis tensor N J by [ , (3.2)]. Thus the tangent space of the Teichm¨ullerspace T ( M ) at the equivalence class of an element J ∈ J int , ( M ) can formally beidentified with the cohomology of the Dolbeault complex (2.2), i.e.(2.3) T [ J ] T ( M ) = ker( ¯ ∂ J : Ω , J ( M, T M ) → Ω , J ( M, T M ))im( ¯ ∂ J : Ω ( M, T M ) → Ω , J ( M, T M )) . The proof requires a local slice theorem for the action of the diffeomorphism groupon the space of integrable complex structures.
ICCI FORM AND TEICHM¨ULLER SPACES 13
For every J ∈ J int , ( M ) the space of holomorphic vector fields is isomorphicto the space of harmonic 1-forms by [ , Lemma 3.9]. Moreover, the Bogomolov–Tian–Todorov theorem asserts that the obstruction class vanishes [
6, 55, 56 ],so the cohomology of the Dolbeault complex (2.2) has constant dimension, andthat T ( M ) is indeed a smooth manifold whose tangent space at the equivalenceclass of J ∈ J int , ( M ) is the cohomology group (2.3). The Teichm¨uller space is ingeneral not Hausdorff, even for the K3 surface (see [
34, 58 ] and also Example 1.6).For hyperK¨ahler manifolds the Teichm¨uller space becomes Hausdorff after identify-ing inseparable complex structures (see Verbitsky [
58, 59 ]) which are biholomorphicby a theorem of Huybrechts [ ].Now fix a positive volume form ρ ∈ Ω n ( M ). Then another description of theTeichm¨uller space of Calabi–Yau structures is as the quotient T ( M, ρ ) := J int , ( M, ρ ) / Diff ( M, ρ ) = J int , ( M, ρ ) / Diff ex ( M, ρ ) , J int , ( M, ρ ) := (cid:8) J ∈ J int , ( M ) | Ric ρ,J = 0 (cid:9) . (2.4)Here the two quotients agree because the quotient group Diff ( M, ρ ) / Diff ex ( M, ρ )acts trivially on J int , ( M, ρ ) / Diff ex ( M, ρ ) (see [ , Lemma 3.9]). The tangentspace of T ( M, ρ ) at the equivalence class of J ∈ J int , ( M, ρ ) is the quotient(2.5) T [ J ] ρ T ( M, ρ ) = (cid:8) b J ∈ Ω , J ( M, T M ) | ¯ ∂ J b J = 0 , d Ric ρ ( J, b J ) = 0 (cid:9)(cid:8) L Y α J | α ∈ Ω n − ( M ) (cid:9) . The inclusion ι ρ : T ( M, ρ ) → T ( M ) is a diffeomorphism by [ , Lemma 4.2]. Theproof uses Moser isotopy and the formula (1.3). The Teichm¨uller space T ( M, ρ ) is asubmanifold of the infinite-dimensional symplectic quotient W ( M, ρ ) in (1.11) andit turns out that the 2-form (1.1) descends to a symplectic form on T ( M, ρ ) andthus induces a symplectic form on T ( M ). An explicit formula for this symplecticform relies on the following two observations. First, it follows from (1.3) that, forevery J ∈ J int , ( M ), there exists a unique positive volume form ρ J ∈ Ω n ( M ) thatsatisfies(2.6) Ric ρ J ,J = 0 , Z M ρ J = V := Z M ρ. Second, for every J ∈ J int , ( M ) and every b J ∈ Ω , J ( M, T M ) with ¯ ∂ J b J = 0, thereexist unique functions f, g ∈ Ω ( M ) that satisfy(2.7) Λ ρ J ( J, b J ) = − df ◦ J + dg, Z M f ρ J = Z M gρ J = 0 . (See [ , Lemma 3.8].) With this understood, define(2.8) Ω WP J ( b J , b J ) := Z M (cid:16) trace (cid:0) b J J b J (cid:1) − f g + f g (cid:17) ρ J for J ∈ J int , ( M ) and b J i ∈ Ω , J ( M, T M ) with ¯ ∂ J b J i = 0, where f i , g i are as in (2.7). Theorem ]) . Equation (2.8) defines a closed -form Ω WP on J int , ( M ) that descends to a symplectic form, still denoted by Ω WP , on the Teichm¨ullerspace T ( M ) . Its pullback under the diffeomorphism ι ρ : T ( M, ρ ) → T ( M ) isthe symplectic form induced by (1.1) and renders T ( M, ρ ) into a symplectic sub-manifold of the infinite-dimensional symplectic quotient W ( M, ρ ) in (1.11) . Here is why this result is not quite as obvious as it may seem at first glance.The space J ( M ) admits a complex structure b J
7→ − J b J and the symplectic form Ω ρ in equation (1.1) is a (1 , M ) >
2, it is not a K¨ahler form because the symmetricbilinear form Ω ρ,J ( b J , − J b J ) = Z M trace( b J b J ) ρ is indefinite on Ω , J ( M, T M ). Thus a complex submanifold of J ( M ) need notbe symplectic. This is precisely the case for the submanifold J int , ( M ) in (2.1),because the restriction of the 2-form Ω ρ,J to T J J int , ( M ) = ker ¯ ∂ J has a nontrivialkernel in the case dim( M ) >
2, which in the case Ric ρ,J = 0 consists of all infinites-imal deformations b J = L X J of complex structures such that both X and JX aredivergence-free. A key ingredient in the proof of this assertion is the fact that thespace of all ¯ ∂ J -harmonic (0 , b J ∈ Ω , J ( M, T M ) on a closed Ricci-flat K¨ahlermanifold is invariant under the homomorphism b J b J ∗ (see [ , Lemma 3.10]). Itthen follows that the kernel of the 2-form Ω WP J in (2.8) on T J J int , ( M ) = ker ¯ ∂ J isthe image of ¯ ∂ J and hence Ω WP J descends to a nondegenerate 2-form on the tangentspace T [ J ] T ( M ) = ker ¯ ∂ J / im ¯ ∂ J for each J ∈ J int , ( M ). This shows that Ω WP descends to a symplectic form on T ( M ) (see [ , Theorem 4.4]).Theorem 2.1 gives an alternative construction of the Weil–Petersson symplecticform on the Teichm¨uller space of Calabi–Yau structures (see [
38, 40, 46, 50, 55,56 ] for the polarized case and [ , Ch 16] for the K3 surface). The Teichm¨ullerspace T ( M ) carries a natural complex structure[ b J ] [ − J b J ]and the Weil–Petersson symplecic form Ω WP is of type (1 , h , and the total dimension is the Hodge number h n − , ( M, L ),where L = Λ n, J T M . These Hodge numbers are deformation invariant by [ ,Proposition 9.30].If a ∈ H ( M ; R ) is a K¨ahler class, then the tangent spaces of the polarizedTeichm¨uller space T ,a ( M ) ⊂ T ( M ) in Remark 2.6 below are positive subspacesfor the Weil–Petersson symplectic form and so Ω WP restricts to a K¨ahler formon T ,a ( M ). If h , = 0, then T ( M ) is Hausdorff and K¨ahler and each polarizedspace T ,a ( M ) is an open subset of T ( M ). Here is a list of the real dimensionsfor the 2 n -torus, the K3 surface, the Enriques surface, the quintic in C P , and thebanana manifold B in [ ]. The last column lists the dimensions of the K¨ahler cones. M T ( M ) T ,a ( M ) K J h n − , ( M, L ) 2 h n − , ( M, L ) − h , h , h , T n n n + n n − n n K3
40 38 2 20Enriques 20 20 0 10Quintic 202 202 0 1 B
16 16 0 20
ICCI FORM AND TEICHM¨ULLER SPACES 15
Let M be aclosed connected oriented 2 n -manifold, let c ∈ H ( M ; R ) be a nonzero cohomologyclass that admits an integral lift, and let ~ be a real number such that (2 π ~ c ) n > J int ,c ( M ) := (cid:8) J ∈ J int ( M ) (cid:12)(cid:12) c R ( J ) = c, π ~ c ∈ K J (cid:9) of all complex structures J on M whose real first Chern class is c and whose K¨ahlercone K J contains the cohomology class 2 π ~ c . For J ∈ J int ,c ( M ) denote by(2.9) S J := (cid:8) ω ∈ Ω ( M ) | dω = 0 , ω n > , [ ω ] = 2 π ~ c, J ∈ J int ( M, ω ) (cid:9) the space of all symplectic forms ω on M that are compatible with J and representthe cohomology class 2 π ~ c . By the Calabi–Yau Theorem [
10, 62 ] each volumeform ρ ∈ Ω n ( M ) with R M ρ = h (2 π ~ c ) n / n ! , [ M ] i and each J ∈ J int ,c ( M ) determinea unique symplectic form ω ρ,J ∈ S J whose volume form is ρ , i.e.(2.10) ω n ρ,J / n ! = ρ. In the case of general type with ~ <
63, 64 ] asserts that everycomplex structure J ∈ J int ,c ( M ) admits a unique symplectic form ω ∈ S J thatsatisfies the K¨ahler–Einstein conditionRic ω n / n ! ,J = ω/ ~ . In the Fano case with ~ >
15, 16 ] assertsthat a complex structure J ∈ J int ,c ( M ) admits a symplectic form ω ∈ S J thatsatisfies the K¨ahler–Einstein condition Ric ω n / n ! ,J = ω/ ~ if and only if it satisfiesthe K-polystablility condition of Yau–Tian–Donaldson [
24, 54, 64 ]. The “only if ” statement was proved earlier by Berman [ ]. Moreover, it was shown by Berman–Berndtsson [ ] that, if ω, ω ′ ∈ S J both satisfy the K¨ahler–Einstein condition, thenthere exists a holomorphic diffeomorphism ψ ∈ Aut ( M, J ) such that ω ′ = ψ ∗ ω .Fix a volume form ρ ∈ Ω n ( M ) with R M ρ = h (2 π ~ c ) n / n ! , [ M ] i and consider thefollowing models for the Teichm¨uller space of K¨ahler–Einstein structures : T c ( M, ρ ) := J KE ,c ( M, ρ ) / Diff ( M, ρ ) , J KE ,c ( M, ρ ) := { J ∈ J int ,c ( M ) | Ric ρ,J = ω ρ,J / ~ } , T c ( M ) := J KE ,c ( M ) / Diff ( M ) , J KE ,c ( M ) := { J ∈ J int ,c ( M ) | J is K-polystable } . (2.11)In general these spaces may be singular. At the equivalence class of a regularelement J ∈ J KE ,c ( M, ρ ), respectively J ∈ J KE ,c ( M ), the tangent spaces are T [ J ] ρ T c ( M, ρ ) = n b J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) ¯ ∂ J b J = 0 , d Ric ρ ( J, b J ) ∧ ω n − ρ,J = 0 o {L X J | X ∈ Vect( M ) , dι ( X ) ρ = 0 } ,T [ J ] T c ( M ) = n b J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) ¯ ∂ J b J = 0 o(cid:8) L X J (cid:12)(cid:12) X ∈ Vect( M ) (cid:9) . (2.12)Combining the theorems of Yau, Berman–Berndtsson, and Chen–Donaldson–Sunwith Moser isotopy, we find that the natural map ι ρ : T c ( M, ρ ) → T c ( M ) is a bijec-tion (and a diffeomorphism on the smooth part). Thus T c ( M, ρ ) inherits the com-plex structure from T c ( M ), and T c ( M ) inherits the symplectic form from T c ( M, ρ ). Lemma
Hodge decomposition) . Let J ∈ J KE ,c ( M ) , choose ω ∈ S J with Ric ω n / n ! ,J = ω/ ~ , define ρ := ω n / n ! , and let b J ∈ Ω , J ( M, T M ) with ¯ ∂ J b J = 0 .Then there exist X ∈ Vect( M ) , F, G ∈ Ω ( M ) , and A ∈ Ω , J ( M, T M ) such that (2.13) b J = L X J + L v F J + L Jv G J + A, (2.14) dι ( X ) ρ = dι ( JX ) ρ = 0 , A = A ∗ , ¯ ∂ J A = 0 , ¯ ∂ ∗ J A = 0 . Thus Λ ρ ( J, A ) = 0 by Remark 1.3. Moreover, X and A are uniquely determinedby b J , the four summands in (2.13) are pairwise L orthogonal, and Λ ρ ( J, b J ) = ~ ι ( X ) ω + d (cid:0) ~ F − d ∗ dF (cid:1) − d (cid:0) ~ G − d ∗ dG (cid:1) ◦ J. (2.15) Proof.
The proof has five steps.
Step 1. If F, G ∈ Ω ( M ) and X ∈ Vect( M ) satisfies dι ( X ) ρ = dι ( JX ) ρ = 0 , then Λ ρ ( J, L X J + L v F J + L Jv G J ) = ~ ι ( X ) ω + d (cid:0) ~ F − d ∗ dF (cid:1) − d (cid:0) ~ G − d ∗ dG (cid:1) ◦ J. This follows from (1.9) with f v F = 0 and f Jv F = − d ∗ dF . Step 2.
There exist functions Φ , Ψ ∈ Ω ( M ) and vector fields X, Y ∈ Vect( M ) such that dι ( X ) ρ = dι ( JX ) ρ = 0 and (2.16) Λ ρ ( J, b J ) = ~ ι ( Y ) ω = ~ (cid:0) ι ( X ) ω + d Φ − d Ψ ◦ J (cid:1) , Y = X + v Φ + Jv Ψ . Choose Y ∈ Vect( M ) such that ι ( Y ) ω = ~ Λ ρ ( J, b J ) and then choose Φ , Ψ ∈ Ω ( M )and X ∈ Vect( M ) such that dι ( X ) ρ = dι ( JX ) ρ = 0 and Y = X + v Φ + Jv Ψ . Step 3.
Let H ∈ Ω ( M ) . Then d ( d ∗ dH − ~ H ) = 0 if and only if L v H J = 0 . We have ι ( Jv H ) ρ = ∗ dH and Λ ρ ( J, L v H J ) = d (cid:0) ~ H − d ∗ dH (cid:1) by (1.9). Hence kL v H J k = − Z M Λ ρ ( J, L v H J ) ∧ ι ( Jv H ) ρ = Z M (cid:0) d ∗ dH − ~ H (cid:1) ( d ∗ dH ) ρ (2.17)by (1.8) and this proves Step 3. Step 4.
Let H ∈ Ω ( M ) such that d ∗ dH = ~ H . Then R M Φ Hρ = R M Ψ Hρ = 0 . By Step 3 we have L v H J = 0 and L Jv H J = 0. Hence, by (1.8) and (2.16),0 = ~ Z M Λ ρ ( J, b J ) ∧ ι ( Jv H ) ρ = Z M d Φ ∧ ι ( Jv H ) ρ = Z M Φ( d ∗ dH ) ρ = ~ Z M Φ Hρ, − ~ Z M Λ ρ ( J, b J ) ∧ ι ( v H ) ρ = Z M d Ψ ∧ ι ( Jv H ) ρ = Z M Ψ( d ∗ dH ) ρ = ~ Z M Ψ Hρ.
This proves Step 4.
Step 5.
We prove the lemma.
By Step 4 there exist
F, G ∈ Ω ( M ) such that ~ F − d ∗ dF = ~ Φ, ~ G − d ∗ dG = ~ Ψ.Thus Z := X + v F + Jv G satisfies Λ ρ ( J, L Z J ) = Λ ρ ( J, b J ) by Step 1 and Step 2,and so (2.13) and (2.15) hold with A := b J − L Z J . Let b ω := dι ( Y ) ω = ~ d Ric ρ ( J, b J ).Then, since ¯ ∂ J b J = 0, it follows from [ , Lemma 3.6] that ω ( b Ju, Jv ) + ω ( Ju, b Jv ) = b ω ( u, v ) − b ω ( Ju, Jv ) = ω (( L Y J ) u, Jv ) + ω ( Ju, ( L Y J ) v )for all u, v ∈ Vect( M ). Hence the endomorphism A = ( b J − L Y J ) + L Y − Z J of T M is symmetric. This proves (2.14) and Lemma 2.2. (cid:3)
ICCI FORM AND TEICHM¨ULLER SPACES 17
Theorem
Weil–Petersson symplectic form) . Let J ∈ J KE ,c ( M ) andlet ω, ρ, b J i , X i , F i , G i , A i for i = 1 , be as in Lemma 2.2. Then Ω WP J (cid:0) b J , b J (cid:1) := Z M trace( A JA ) ρ = Z M trace( b J J b J ) ρ − ~ Z M ω ρ,J ( X , X ) ρ + Z M (cid:16)(cid:0) d ∗ dF − ~ F (cid:1) d ∗ dG − (cid:0) d ∗ dG − ~ G (cid:1) d ∗ dF (cid:17) ρ. (2.18) The formula (2.18) defines a K¨ahler form on the Teichm¨uller space T c ( M ) and themapping class group Diff c ( M ) / Diff ( M ) of isotopy classes of diffeomorphisms thatpreserve the cohomology class c acts on T c ( M ) by K¨ahler isometries. The pullbackof Ω WP to T c ( M, ρ ) is the -form Ω WP ρ given by Ω WP ρ,J (cid:0) b J , b J (cid:1) = Z M trace( b J J b J ) ρ − ~ Z M Λ ρ ( J, b J ) ∧ Λ ρ ( J, b J ) ∧ ω n − ρ,J ( n − for J ∈ J KE ,c ( M, ρ ) and b J i ∈ Ω , J ( M, T M ) with ¯ ∂ J b J i = 0 , d Ric ρ ( J, b J i ) ∧ ω n − ρ,J = 0 . Proof.
The second equality in (2.18) follows from (2.17) and the fact that theterms L X J , L v F J , L Jv G J , A in Lemma 2.2 are pairwise L orthogonal. Also, foreach J ∈ J KE ,c ( M ), it follows directly from the definition that the skew-symmetricbilinear form Ω WP J on the kernel of ¯ ∂ J in Ω , J ( M, T M ) descends to a nondegenerateform on the quotient space T [ J ] T c ( M ) = ker ¯ ∂ J / im ¯ ∂ J that is compatible with thelinear complex structure [ b J ] [ − J b J ]. That the 2-form Ω WP descends to T c ( M )and is preserved by the action of the mapping class group follows from the nat-urality of the decomposition in Lemma 2.2. Its pullback to T c ( M, ρ ) is givenby (2.19), because d Ric ρ ( J, b J ) ∧ ω n − = 0 implies d ( ~ G − d ∗ dG ) = 0 in Lemma 2.2.That this pullback is closed follows from the equations ∂ t ω ρ,J = ~ d Λ ρ ( J, ∂ t J )and ∂ s Λ ρ ( J, ∂ t J ) − ∂ t Λ ρ ( J, ∂ s J ) = − d trace(( ∂ s J ) J ( ∂ t J )) (in [ , Theorem 2.7])for every smooth map R → J KE ,c ( M, ρ ) : ( s, t ) J s,t . (cid:3) The space K ( M, a, ρ ) of all K¨ahler pairs ( ω, J ) that satisfy [ ω ] = a := 2 π ~ c and ω n / n ! = ρ is not a symplectic submanifold of P ( M, a, ρ ) whenever dim( M ) > K ( M, a, ρ ) at every K¨ahler–Einstein pair ( ω, J ) ∈ K ( M, a, ρ )with Ric ρ,J = ω/ ~ is the set of all pairs ( L X ω, L X J ) such that both X and JX aredivergence-free. Nevertheless, if H ( M ; R ) = 0, then Theorem 2.3 shows that theregular part of T c ( M, ρ ) embeds via [ J ] [ ω ρ,J , J ] as a symplectic submanifoldinto the symplectic quotient W SE ( M, a, ρ ) in (1.17). The condition H ( M ; R ) = 0is necessarily satisfied in the Fano case ~ >
0. In the case ~ < ( M )acts on J KE ,c ( M ) = J int ,c ( M ) with finite isotropy, the quotient Z c ( M, ρ ) := J KE ,c ( M, ρ ) / Diff ex ( M, ρ )embeds symplectically into W SE ( M, a, ρ ) via [ J ] [ ω ρ,J , J ], and the space Z c ( M, ρ )fibers over T c ( M, ρ ) with symplectic fibers. If the action of the group Diff ( M, ρ )on J KE ,c ( M, ρ ) is free, then each fiber is isomorphic to H n − ( M ; R ) / Γ ρ , where Γ ρ is the image of the flux homomorphism Flux ρ : π (Diff ( M, ρ )) → H n − ( M ; R ). Let (
M, ω ) be a closedconnected symplectic 2 n -manifold with the volume form ρ := ω n / n ! and denoteby J int ( M, ω ) the space of all complex structures on M that are compatible with ω .This space is connected for rational and ruled surfaces [ ], however, in general this isan open question. The regular part of J int ( M, ω ) is an infinite-dimensional K¨ahlersubmanifold of J ( M, ω ) whose tangent space at a regular element J is T J J int ( M, ω ) = n J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) b J = b J ∗ , ¯ ∂ J b J = 0 o . Consider the symplectic quotient Z ( M, ω ) := J cscK ( M, ω ) / Ham(
M, ω ) , J cscK ( M, ω ) := (cid:8) J ∈ J int ( M, ω ) | S ω,J = c ω (cid:9) . (2.20)The regular part of this space is a complex submanifold of the infinite-dimensionalquotient W csc ( M, ω ) in (1.25) with the tangent spaces T [ J ] ω Z ( M, ω ) = n b J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) ¯ ∂ J b J = 0 , b J = b J ∗ , b S ω ( J, b J ) = 0 o(cid:8) L v H J (cid:12)(cid:12) H ∈ Ω ( M ) (cid:9) (2.21)for J ∈ J cscK ( M, ω ). So Z ( M, ω ) inherits the K¨ahler structure of W csc ( M, ω ) withthe symplectic from (1.22) and the complex structure [ b J ] ω [ − J ( b J − L v H J )] ω ,where H ∈ Ω ( M ) satisfies b S ω ( J, J b J − J L v H J ) = 0 as in Proposition 1.16. To de-scribe Z ( M, ω ) as a complex quotient, we digress briefly into GIT.
Remark
Geometric invariant theory) . Let (
X, ω, J ) be a closed K¨ahlermanifold and let G be a compact Lie group which acts on X by K¨ahler isometries.Assume that the Lie algebra g := Lie(G) is equipped with an invariant inner prod-uct and that the action is Hamiltonian and is generated by an equivariant momentmap µ : X → g . Then the complexified group G c acts on X by holomorphic diffeo-morphisms and the symplectic quotient X//
G := µ − (0) / G is naturally isomorphicto the complex quotient X ps / G c of the set X ps := { x ∈ X | G c ( x ) ∩ µ − (0) = ∅} of µ -polystable elements of X by the complexified group. The set of µ -polystableelements can be characterized in terms of Mumford’s numerical invariants (2.22) w µ ( x, ξ ) := lim t →∞ h µ (exp( i tξ ) x ) , ξ i associated to x ∈ X and 0 = ξ ∈ g . The moment-weight inequality asserts that(2.23) sup = ξ ∈ g − w µ ( x, ξ ) | ξ | ≤ inf g ∈ G c | µ ( gx ) | and that equality holds whenever the right hand side is positive. An immediateconsequence is that an element x ∈ X is µ -semistable (i.e. G c ( x ) ∩ µ − (0) = ∅ )if and only if w µ ( x, ξ ) ≥ ξ ∈ g \ { } . For µ -polystability the additionalcondition is required that w µ ( x, ξ ) = 0 implies lim t →∞ exp( i tξ ) x ∈ G c ( x ). This isthe content of the Hilbert–Mumford criterion . Its proof is based on the studyof the gradient flow of the moment map squared f := | µ | : X → R and the relatedgradient flow of the Kempf–Ness function Φ x : G c / G → R , defined by(2.24) d Φ x ( g ) b g = −h µ ( g − x ) , Im( g − b g ) i , Φ x ( u ) = 0for b g ∈ T g G c and u ∈ G. The symmetric space G c / G is a Hadamard space and theKempf–Ness function is convex along geodesics. By the
Kempf–Ness Theorem it is bounded below if and only if x is µ -semistable. For an exposition see [ ]. ICCI FORM AND TEICHM¨ULLER SPACES 19
Remark
The space of K¨ahler potentials) . It was noted by Donaldsonin his landmark paper [ ] that much of geometric invariant theory carries over(in part conjecturally) to the infinite-dimensional setting where X is replaced bythe space J ( M, ω ) and the compact Lie group G by G = Ham( M, ω ). Whilein this situation there is no complexified group there do exist complexified grouporbits. In the integrable case the complexified group orbit of J ∈ J int ( M, ω ) is thespace G c ( J ) of all elements J ′ ∈ J int ( M, ω ) that are exact isotopic to J (i.e.there exists a smooth path [0 , → J int ( M, ω ) : s J s and a smooth family ofvector fields [0 , → Vect(
M, ω ) : s v s such that J = J , J = J ′ , ∂ s J s = L v s J s ,and ι ( v s ) ω = d Φ s − d Ψ s ◦ J s for Φ s , Ψ s ∈ Ω ( M )). In this situation the role of thesymmetric space G c / G is played by the space of K¨ahler potentials (2.25) H J := (cid:26) h ∈ Ω ( M ) (cid:12)(cid:12)(cid:12)(cid:12) The 2-form ω h := ω + d ( dh ◦ J ) = ω − i ∂ ¯ ∂h satisfies ω h ( b x, J b x ) > ∀ = b x ∈ T M (cid:27) . This space has been studied by Calabi, Chen [
9, 11, 13, 14 ], Mabuchi [
41, 42 ],Semmes [ ], Donaldson [ ] and others. It is an infinite-dimensional symmetricspace of nonpositive sectional curvature with the Mabuchi metric(2.26) h b h , b h i h := Z M b h b h ω n h n !and geodesics are the solutions t h t of the Monge–Amp`ere equation (2.27) ∂ t ∂ t h t + | d∂ t h t | h t = 0 . In [ ] Chen proved that any two elements of H J can be joined by a weak C , geodesic. As noted by Donaldson [
19, 20 ], the analogues of Mumford’s numeri-cal invariants in this setting are the
Futaki invariants [ ], the analogue of theKempf–Ness function is the Mabuchi functional [ ] M J : H J → R defined by(2.28) d M J ( h ) b h = Z M (cid:0) S ω h ,J − c ω (cid:1)b h ω n h n ! , M J (0) = 0 , and the analogue of the gradient flow of the moment map squared is the Calabi flow.After earlier results by Donaldson [
21, 22 ], Mabuchi [ ], and Chen–Tian [ ] itwas shown by Berman–Berndtsson [ ] that the Mabuchi functional is convex alongweak geodesics, that every K¨ahler potential h ∈ H J with constant scalar curva-ture S ω h ,J = c ω minimizes the Mabuchi functional, and that constant scalar curva-ture K¨ahler metrics are unique up to holomorphic diffeomorphism, i.e. if two K¨ahlerpotentials h, h ′ ∈ H J have constant scalar curvature S ω h ,J = S ω h ′ ,J = c ω , thenthere exists a holomorphic diffeomorphism ψ ∈ Aut ( M, J ) such that ω h ′ = ψ ∗ ω h .In the present setting the analogue of the Hilbert–Mumford criterion is the Yau–Tian–Donaldson conjecture [
24, 54, 64 ] which relates the existence of a constantscalar curvature K¨ahler potential to K-polystability. For Fano manifolds it wasconfirmed by Chen–Donaldson–Sun [
15, 16 ], while in general it is an open question.Remark 2.5 shows that, according to the YTD conjecture, the space Z ( M, ω )can be expressed as the quotient Z K ( M, ω ) := J K ( M, ω ) / ∼ , where J K ( M, ω ) isthe space of K-polystable complex structures that are compatible with ω , and theequivalence relation is exact isotopy as in Remark 2.5. The formal (Zariski type)tangent space of Z K ( M, ω ) at the equivalence class of an element J ∈ J K ( M, ω ) isthe quotient T [ J ] Z K ( M, ω ) = T J J int ( M, ω ) / {L v F + Jv G J | F, G ∈ Ω ( M ) } and thecomplex structure is [ b J ] [ − J b J ]. It is also of interest to consider the
Teichm¨uller space of constant scalarcurvature K¨ahler metrics , defined by T ( M, ω ) := J cscK ( M, ω ) / Symp ( M, ω ) . (2.29)If H ( M ; R ) = 0, then Ham( M, ω ) = Symp ( M, ω ) and hence the regular part ofthe Teichm¨uller space T ( M, ω ) = Z ( M, ω ) is K¨ahler. The two spaces also agree inthe Calabi–Yau case, where Symp ( M, ω ) / Ham(
M, ω ) acts trivially on Z ( M, ω ).In the K¨ahler–Einstein case with 2 π ~ c R ( ω ) = [ ω ] and ~ < ( M, ω )acts on J cscK ( M, ω ) with finite isotropy, the Teichm¨uller space T ( M, ω ) carries aK¨ahler form given by (2.19) with ω ρ,J = ω , and Z ( M, ω ) fibers over T ( M, ω ) withsymplectic fibers. If the action of Symp ( M, ω ) on J cscK ( M, ω ) is free, then eachfiber is isomorphic to the space H ( M ; R ) / Γ ω , where Γ ω is the image of the fluxhomomorphism Flux ω : π (Symp ( M, ω )) → H ( M ; R ). Remark . Fix a symplectic form ω on M that admits a compatible complexstructure J with S ω,J = c ω and Aut( M, J ) ∩ Diff ( M ) = Aut ( M, J ). Define ρ := ω n / n ! , a = [ ω ] ∈ H ( M ; R ) , c := c R ( ω ) ∈ H ( M ; R ) . Assume the Calabi–Yau case c = 0 and consider the polarized Teichm¨uller space T ,a ( M, ρ ) := { J ∈ J int , ( M ) | Ric ρ,J = 0 , a ∈ K J } / Diff ( M, ρ )(2.30)of all isotopy classes of Ricci-flat complex structures that contain the cohomologyclass a in their K¨ahler cone. This space is Hausdorff and the Weil–Petersson metricon T ,a ( M, ρ ) is K¨ahler. Moreover, there is a natural holomorphic map ι ω : T ( M, ω ) → T ,a ( M, ρ )which pulls back the Weil–Petersson symplectic form on T ,a ( M, ρ ) in Theorem 2.1to the symplectic form (1.22) on T ( M, ω ). The map ι ω need not be injective orsurjective. It is injective if and only if Symp( M, ω ) ∩ Diff ( M ) = Symp ( M, ω ).In [ ] Seidel found many examples of symplectic four-manifolds that admit sym-plectomorphisms that are smoothly, but not symplectically, isotopic to the identity,including K3-surfaces with embedded Lagrangian spheres. The map ι ω is surjectiveif and only if the space S ,a of symplectic forms in the class a with real first Chernclass zero that admit compatible complex structures is connected. By a theorem ofHajduk–Tralle [ ] the space S ,a is disconnected for the 8 k -torus with k ≥ π ~ c = a for some nonzero real number ~ .Then there is again a natural holomorphic map ι ω : T ( M, ω ) → T c ( M, ρ )which pulls back the Weil–Petersson symplectic form on T c ( M, ρ ) in Theorem 2.3 tothe symplectic form (2.19) on T ( M, ω ). As before, this map need not be injectiveor surjective. Seidel’s examples in dimension four include as Fano manifolds the k -fold blowup of the projective plane with 5 ≤ k ≤ ι ω is not injective. It is surjective if andonly if the space S c,a of symplectic forms in the class a with first Chern class c that admit compatible K-polystable complex structures is connected. By a theoremof Randal-Williams [ ] every complete intersection M with dim( M ) = 16 k ≥ H k ( M ; R )) ≥ φ that acts as the identity onhomology such that ω is not isotopic to φ ∗ ω for every symplectic form ω . Thisincludes K¨ahler–Einstein examples, and in these cases S c,a is disconnected. ICCI FORM AND TEICHM¨ULLER SPACES 21
3. Fano manifolds
This section explains how the symplectic form introduced by Donaldson [ ]on the space of Fano complex structures fits into the present setup. We begin bygiving another proof of nondegeneracy, and then discuss Berndtsson convexity forthe Ding functional and the Donaldson–K¨ahler–Ricci flow. Let (
M, ω ) be a closed connectedsymplectic 2 n -manifold that satisfies the Fano condition(3.1) 2 πc R ( ω ) = [ ω ] ∈ H ( M ; R ) . As in Subsection 1.3 we denote by v F the Hamiltonian vector field of F andby { F, G } the Poisson bracket of F, G ∈ Ω ( M ). Let J int ( M, ω ) be the space of ω -compatible complex structures on M . If this space is nonempty, then M is simplyconnected. Throughout we will ignore all regularity issues and treat J int ( M, ω ) asa submanifold of J ( M, ω ) whose tangent space at J ∈ J int ( M, ω ) is T J J int ( M, ω ) = n b J ∈ Ω , J ( M, T M ) (cid:12)(cid:12) ¯ ∂ J b J = 0 , b J = b J ∗ o . This is a complex subspace of T J J ( M, ω ) and so inherits the symplectic form (1.22)from the ambient K¨ahler manifold J ( M, ω ). In [ ] Donaldson introduced anothersymplectic form on J int ( M, ω ) which we explain next.It follows from (1.3) that, for every complex structure J ∈ J int ( M, ω ), thereexists a unique positive volume form ρ J ∈ Ω n ( M ) that satisfies(3.2) Ric ρ J ,J = ω, Z M ρ J = 1 . Moreover, d Ric ρ J ( J, b J ) is an exact (1 , b J ∈ T J J int ( M, ω ) by (3.2)and [ , Lemma 3.6]. Thus, by the ∂ ¯ ∂ -lemma, for every J ∈ J int ( M, ω ) andevery b J ∈ T J J int ( M, ω ), there exist unique functions f, g ∈ Ω ( M ) that satisfy(3.3) Λ ρ J ( J, b J ) = − df ◦ J + dg, Z M f ρ J = Z M gρ J = 0 . The
Donaldson symplectic form on J int ( M, ω ) is defined by(3.4) Ω DJ ( b J , b J ) := Z M (cid:16) trace (cid:0) b J J b J (cid:1) − f g + g f (cid:17) ρ J for J ∈ J int ( M, ω ) and b J i ∈ T J J int ( M, ω ), where ρ J , f i , g i are as in (3.2) and (3.3).The fact that the 2-form (3.4) is nondegenerate is far from trivial and is one ofthe main results in [ ]. Indeed, as noted by Donaldson, nondegeneracy can beviewed as a reformulation of Berndtsson’s convexity theorem [
4, 5 ] for the Dingfunctional [ ] on the space of K¨ahler potentials (see Subsection 3.2 below).The symplectic form Ω D in (3.4) is given by essentially the same formula asthe Weil–Petersson symplectic form Ω WP on T ( M ) in (2.8). In contrast to theCalabi–Yau case, where the lifted 2-form on J int , ( M ) has the kernel im ¯ ∂ J on eachtangent space T J J int , ( M ), the 2-form Ω D is nondegenerate in the Fano case.The definition of the symplectic form in Donaldson’s paper [ ] uses the exis-tence of a holomorphic n -form with values in a suitable holomorphic line bundle todefine the volume form denoted by ρ J in (3.2). That the 2-form (3.4) agrees withthe symplectic form in [ ] (up to a factor 1 /
4) then follows from the discussionin [ , Appendix D]. Theorem
Donaldson [ ] ) . Ω D is a Symp(
M, ω ) -invariant symplec-tic form on J int ( M, ω ) and is compatible with the complex structure b J
7→ − J b J .The action of Ham(
M, ω ) on J int ( M, ω ) is Hamiltonian and is generated by the Symp(
M, ω ) -equivariant moment map µ : J int ( M, ω ) → (Ω ( M ) / R ) ∗ , given by (3.5) h µ ( J ) , H i := 2 Z M H (cid:18) V ω n n ! − ρ J (cid:19) , V := Z M ω n n ! , for J ∈ J int ( M, ω ) and H ∈ Ω ( M ) , where ρ J ∈ Ω n ( M ) is as in (3.2) . Thus (3.6) Ω DJ ( b J, L v H J ) = − Z M Hf ρ J = h dµ ( J ) b J , H i for b J ∈ T J J int ( M, ω ) and H ∈ Ω ( M ) , where f is as in (3.3) . Remark
Weil–Petersson symplectic form) . The zero set of the mo-ment map in Theorem 3.1 is the space J KE ( M, ω ) := (cid:8) J ∈ J int ( M, ω ) | Ric ω n / n ! ,J = ω (cid:9) = J cscK ( M, ω )of K¨ahler–Einstein complex structures compatible with ω . Since M is simplyconnected, the quotient T KE ( M, ω ) := J KE ( M, ω ) / Ham(
M, ω ) = T ( M, ω ) is theTeichm¨uller space in (2.29) and the symplectic form on this space induced by (3.4)is 1 /V times the Weil–Petersson symplectic form induced by (1.22). For the relationto the Teichm¨uller space T c ( M, ρ ) ∼ = T c ( M ) in Theorem 2.3 see Remark 2.6.Below we give a proof of nondegeneracy of (3.4) which amounts to translatingthe argument in [ ] into our notation. The heart of the proof is Lemma 3.7. Definition . Fix a complex structure J ∈ J int ( M, ω ) and let ρ := ω n / n !.The K¨ahler–Ricci potential of J is the function Θ ω,J := Θ J : M → (0 , ∞ )defined by Θ J := ρ J /ρ . Hence Ric Θ J ρ,J = ω by (3.2) and so by (1.3), we have(3.7) d ( d log(Θ J ) ◦ J ) = ω − Ric ρ,J , Z M Θ J ρ = 1 . Thus Θ J = 1 /V if and only if ( M, ω, J ) is a K¨ahler–Einstein manifold. Denoteby d ∗ d : Ω ( M ) → Ω ( M ) the Laplace–Beltrami operator of the Riemannian met-ric h· , ·i := ω ( · , J · ) and define the linear operators L , B : Ω ( M ) → Ω ( M ) by(3.8) L F := d ∗ dF − h v Θ J , v F i Θ J , B F := { Θ J , F } Θ J for F ∈ Ω ( M ). Thus L is a self-adjoint Fredholm operator and B is skew-adjointwith respect to the L inner product h F, G i J := R M F Gρ J on Ω ( M ). Lemma
K¨ahler–Ricci potential) . Choose elements J ∈ J int ( M, ω ) and b J ∈ T J J int ( M, ω ) and let f and Θ J be as in (3.3) and (3.7) . Then (3.9) b Θ ω ( J, b J ) := ∂∂t (cid:12)(cid:12) t =0 Θ J t = f Θ J for every smooth path R → J int ( M, ω ) : t J t with J = J and ∂∂t (cid:12)(cid:12) t =0 J t = b J . Proof.
By Proposition 1.2 and (1.3) the derivative of any path t Ric ρ t ,J t is given by ∂ t Ric ρ t ,J t = d Ric ρ t ( J t , ∂ t J t ) + d ( d ( ∂ t ρ t /ρ t ) ◦ J t ). In the case at handwith J t ∈ J int ( M, ω ) and ρ t = ρ J t = Θ J t ω n / n ! this yields the equation0 = d Ric ρ J ( J, b J ) + d (cid:0) d (cid:0) b Θ ω ( J, b J ) / Θ J (cid:1) ◦ J (cid:1) = d (cid:0) d (cid:0) b Θ ω ( J, b J ) / Θ J − f (cid:1) ◦ J (cid:1) . Since b Θ ω ( J, b J ) / Θ J − f has mean value zero for ρ J , this proves the lemma. (cid:3) ICCI FORM AND TEICHM¨ULLER SPACES 23
Lemma
Holomorphic vector fields) . Let J ∈ J int ( M, ω ) , choose thequadruple ρ J , Θ J , L , B as in Definition 3.3, and choose functions F, G ∈ Ω ( M ) such that R M F ρ J = R M Gρ J = 0 . Then the following holds. (i) F = G = 0 if and only if L F + B G = 0 and L G − B F = 0 . (ii) Λ ρ J ( J, L v F + Jv G J ) = − d (2 G − L G + B F ) ◦ J + d (2 F − L F − B G ) . (iii) L v F + Jv G J = 0 if and only if L F + B G = 2 F and L G − B F = 2 G . Proof.
Throughout the proof we use the notation (cid:13)(cid:13) F (cid:13)(cid:13) := sZ M F ρ J , (cid:13)(cid:13) v (cid:13)(cid:13) := sZ M ω ( v, Jv ) ρ J , (cid:13)(cid:13) b J (cid:13)(cid:13) := sZ M trace( b J ) ρ J for F ∈ Ω ( M ), v ∈ Vect( M ), and b J ∈ Ω , J ( M, T M ) with b J = b J ∗ .To prove part (i), we observe that Z M ( L F ) Gρ J = Z M ( dF ◦ J ) ∧ dG ∧ Θ J ω n − ( n − Z M ω ( v F , Jv G ) ρ J , Z M ( B F ) Gρ J = Z M dF ∧ dG ∧ Θ J ω n − ( n − Z M ω ( v F , v G ) ρ J (3.10)for all F, G ∈ Ω ( M ), and hence(3.11) k v F + Jv G k = Z M (cid:16) F ( L F + B G ) + G ( L G − B F ) (cid:17) ρ J . Now let
F, G ∈ Ω ( M ) such that L F + B G = L G − B F = 0. Then v F + Jv G = 0by (3.11) and hence R M ω ( v F , Jv F ) ω n / n ! = R M ω ( v F , v G ) ω n / n ! = 0. Thus F and G are constant and this proves (i). Part (ii) follows from (1.9), (3.2), and(3.12) dι ( v F ) ρ J = ( B F ) ρ J , dι ( Jv G ) ρ J = − ( L G ) ρ J . Moreover, kL v F + Jv G J k = − R M Λ ρ J ( J, L v F + Jv G J ) ∧ ι ( J ( v F + Jv G )) ρ J by (1.8),and hence (iii) follows from (ii). (cid:3) Lemma
Decomposition Lemma) . Let J ∈ J int ( M, ω ) , let ρ J be asin (3.2) , and let b J ∈ Ω , J ( M, T M ) such that ¯ ∂ J b J = 0 and b J = b J ∗ with respect tothe metric ω ( · , J · ) . Then there exist F, G ∈ Ω ( M ) and A ∈ Ω , J ( M, T M ) such that (3.13) b J = L v F J + L Jv G J + A, and Z M F ρ J = Z M Gρ J = 0 , A = A ∗ , ¯ ∂ J A = 0 , Λ ρ J ( J, A ) = 0 . (3.14) Moreover, A and L v F + Jv G J are L orthogonal If b J satisfies (3.13) and (3.14) ,then Λ ρ J ( J, b J ) is given by (3.3) with (3.15) f = 2 G − L G + B F, g = 2 F − L F − B G, and b J satisfies the equation Z M (cid:16) trace (cid:0) b J (cid:1) − f − g (cid:17) ρ J = Z M trace (cid:0) A (cid:1) ρ J + 2 Z M (cid:16) | v F + Jv G | − (cid:0) F + G (cid:1)(cid:17) ρ J . (3.16) Proof.
Let f, g ∈ Ω ( M ) be as in (3.3) We prove first that all F, G ∈ Ω ( M )satisfies the identity(3.17) Z M trace( b J L v F + Jv G J ) ρ J = − Z M (cid:16) g ( L F + B G ) + f ( L G − B F ) (cid:17) ρ J . To see this, abbreviate v := v F + Jv G . Then we have dι ( v ) ρ J = ( − L G + B F ) ρ J and dι ( Jv ) ρ J = ( − L F − B G ) ρ J by (3.12). Hence, by (1.8) and (3.3), Z M trace (cid:0) b J L v J (cid:1) ρ J = Z M (cid:16) df ◦ J − dg (cid:17) ∧ ι ( Jv ) ρ J = Z M (cid:16) f dι ( v ) ρ J + gdι ( Jv ) ρ J (cid:17) = Z M (cid:16) f ( − L G + B F ) + g ( − L F − B G ) (cid:17) ρ J . This proves (3.17).Now choose functions
F, G ∈ Ω ( M ) such that L F + B G = 2 F, L G − B F = 2 G. Then L v F + Jv G J = 0 by part (iii) of Lemma 3.5 and hence R M ( gF + f G ) ρ J = 0by equation (3.17). Thus the pair ( g, f ) is L orthogonal to the kernel of the self-adjoint Fredholm operator ( F, G ) (2 F − L F − B G, G − L G + B F ) and so belongsto its image. Hence there exist smooth functions F, G ∈ Ω ( M ) such that2 F − L F − B G = g, G − L G + B F = f. By part (ii) of Lemma 3.5 this implies Λ ρ J ( J, b J − L v F + Jv G J ) = 0 . Since L v F J and L Jv G J = J L v G J are symmetric, by [ , Lemma 3.7], this proves (3.14).Now assume that F, G, A have been found such that b J satisfies (3.13) and (3.14).Then (3.15) follows directly from part (ii) of Lemma 3.5. Moreover, Z M trace( A L v F + Jv G J ) ρ J = 0by (1.8) and (3.14). Hence, by (3.17) we have Z M trace (cid:0) b J (cid:1) ρ J − Z M trace (cid:0) A (cid:1) ρ J = Z M trace (cid:0) b J ( L v F + Jv G J ) (cid:1) ρ J = Z M (cid:16) f ( − L G + B F ) + g ( − L F − B G ) (cid:17) ρ J = Z M (cid:16) f ( f − G ) + g ( g − F ) (cid:17) ρ J = Z M (cid:16) f + g (cid:17) ρ J + 2 Z M (cid:16) F ( L F + B G − F ) + G ( L G − B F − G ) (cid:17) ρ J = Z M (cid:0) f + g (cid:1) ρ J + 2 Z M (cid:16) | v F + Jv G | − (cid:0) F + G (cid:1)(cid:17) ρ J . Here the last step uses (3.11). This proves (3.16). (cid:3)
ICCI FORM AND TEICHM¨ULLER SPACES 25
Lemma
Berndtsson Inequality) . Let J ∈ J int ( M, ω ) , choose ρ J asin (3.2) , and let F, G ∈ Ω ( M ) such that R M F ρ J = R M Gρ J = 0 . Then (3.18) Z M | v F + Jv G | ρ J ≥ Z M (cid:0) F + G (cid:1) ρ J , and equality holds in (3.18) if and only if L v F + Jv G J = 0 . If L v F + Jv G J = 0 , thenevery pair of functions b F , b G ∈ Ω ( M ) with R M b F ρ J = R M b Gρ J = 0 satisfies (3.19) Z M (cid:10) v F + Jv G , v b F + Jv b G (cid:11) = 2 Z M (cid:16) F b F + G b G (cid:17) ρ J . Proof.
Since F and G have mean value zero, it follows from part (i) ofLemma 3.5 that there exists a unique pair of functions Φ , Ψ ∈ Ω ( M ) such that(3.20) L Φ + B Ψ = F, L Ψ − B Φ = G, Z M Φ ρ J = Z M Ψ ρ J = 0 . Continue the notation in the proof of Lemma 3.5 and define(3.21) u := v Φ + Jv Ψ , v := v F + Jv G . Then Lemma 3.6 with b J = L u J , f = 2Ψ − G , g = 2Φ − F , A = 0 yields kL u J k = Z M (cid:16) | u | − (cid:0) Φ + Ψ (cid:1) + (cid:0) − F (cid:1) + (cid:0) − G (cid:1) (cid:17) ρ J = Z M (cid:16) | u | + F + G − F − G (cid:17) ρ J = Z M (cid:16) F + G (cid:17) ρ J − k u k . (3.22)The last step uses the formula k u k = R M (Φ F + Ψ G ) ρ J in (3.11). Now, for λ ∈ R , k v k − k v − λu k = 2 λ Z M ω ( u, Jv ) ρ J − λ k u k = 2 λ Z M (cid:16) ω ( v Φ , Jv F ) + ω ( v Ψ , v F ) + ω ( v Ψ , Jv G ) − ω ( v Φ , v G ) (cid:17) ρ J − λ k u k = 2 λ Z M (cid:16) ( L Φ + B Ψ) F + ( L Ψ − B Φ) G (cid:17) ρ J − λ k u k = 2 λ Z M (cid:0) F + G (cid:1) ρ J − λ k u k = (cid:18) λ − λ (cid:19) Z M (cid:0) F + G (cid:1) ρ J + λ kL u J k . Here the second equality follows from (3.21), the third from (3.10), the fourthfrom (3.20), and the last from (3.22). With λ = 2 this yields(3.23) k v k − Z M (cid:0) F + G (cid:1) ρ J = kL u J k + k v − u k ≥ . This proves (3.18). Moreover, equality in (3.18) implies v = 2 u and L u J = 0, andso L v J = 0. Conversely, if L v J = 0, then L F + B G = 2 F and L G − B F = 2 G bypart (iii) of Lemma 3.5, hence the unique solution of (3.20) is given by Φ = F and Ψ = G , which implies u = v and L u J = 0, so equality in (3.18) followsfrom (3.23). To prove the last assertion, define F t := F + t b F and G t := G + t b G and differentiate the function t R M (cid:0) | v F t + Jv G t | − F t − G t (cid:1) ρ J at t = 0. (cid:3) Proof of Theorem 3.1.
Fix an element J ∈ J int ( M, ω ) and let ρ J be asin (3.2). We show first that the 2-form (3.4) is nondegenerate and compatible withthe complex structure b J
7→ − J b J . To see this, let b J ∈ T J J int ( M, ω ), let f, g be asin (3.3), and let
F, G, A be as in Lemma 3.6. Then (3.4) and (3.16) yieldΩ DJ ( b J, − J b J ) = Z M (cid:16) trace (cid:0) b J (cid:1) − f − g (cid:17) ρ J = Z M trace (cid:0) A (cid:1) ρ J + 2 Z M (cid:16) | v F + Jv G | − (cid:0) F + G (cid:1)(cid:17) ρ J . (3.24)By Lemma 3.7 the right hand side in (3.24) is nonnegative and vanishes if and onlyif A = 0 and L v F + Jv G J = 0 or, equivalently, b J = 0. This proves nondegeneracy.To prove (3.6), fix an element b J ∈ T J J int ( M, ω ), let f, g be as in (3.3), andlet H ∈ Ω ( M ) such that R M Hρ J = 0. Then, by Lemma 3.5 and (3.10), we haveΩ DJ ( b J , L v H J ) = Z M trace (cid:0) b JJ L v H J (cid:1) ρ J − Z M f (2 H − L H ) ρ J + Z M g ( B H ) ρ J = Z M Λ ρ J ( J, b J ) ∧ ι ( v H ) ρ J + Z M dH ∧ dg ∧ Θ J ω n − ( n − Z M ( df ◦ J ) ∧ dH ∧ Θ J ω n − ( n − − Z M Hf ρ J = − Z M Hf ρ J . Here the last equality holds because Λ ρ J ( J, b J ) = − df ◦ J + dg . This proves the firstequality in (3.6) and the second follows from Lemma 3.4.It remains to prove that the 2-form (3.4) is closed. In Donaldson’s formulationthis follows directly from the definition, while in our formulation this requires proof.Here is the outline. First, let R → J int ( M, ω ) : ( s, t ) J s,t be a smooth map and,for s, t ∈ R , define the functions f s , g s , f t , g t ∈ Ω ( M ) such that they have meanvalue zero with respect to ρ J for J = J s,t andΛ ρ J ( J, ∂ s J ) = − df s ◦ J + dg s , Λ ρ J ( J, ∂ t J ) = − df t ◦ J + dg t . Here we have dropped the subscripts s, t for J and observe that f s , g s , f t , g t dependalso on s and t . Then by [ , Theorem 2.7] and Lemma 3.4 we have ∂ s Λ ρ J ( J, ∂ t J ) − ∂ t Λ ρ J ( J, ∂ s J ) + d trace (cid:0) ( ∂ s J ) J ( ∂ t J ) (cid:1) = df s ◦ ∂ t J − df t ◦ ∂ s J. Hence a calculation shows that ∂ s f t − ∂ t f s = 0 , d (cid:16) ∂ s g t − ∂ t g s + trace (cid:0) ( ∂ s J ) J ( ∂ t J ) (cid:1)(cid:17) = 0 . (3.25)Now let R → J int ( M, ω ) : ( r, s, t ) J ( r, s, t ) be a smooth map and define thefunctions f r , f s , f t , g r , g s , g t as before. Then ∂ r ρ J = f r ρ J by Lemma 3.4 and hence ∂ r Ω DJ ( ∂ s J, ∂ t J ) = Z M f r trace (cid:0) ( ∂ s J ) J ( ∂ t J ) (cid:1) ρ J + Z M trace (cid:0) ( ∂ r ∂ s J ) J ( ∂ t J ) (cid:1) ρ J + Z M trace (cid:0) ( ∂ s J ) J ( ∂ r ∂ t J ) (cid:1) ρ J + Z M (cid:16) ( ∂ r g s ) f t + g s ( ∂ r f t ) + f r g s f t − ( ∂ r f s ) g t − f s ( ∂ r g t ) − f r f s g t (cid:17) ρ J . Take a cyclic sum and use (3.25) to obtain ( d Ω D ) J ( ∂ r J, ∂ s J, ∂ t J ) = 0. (cid:3) ICCI FORM AND TEICHM¨ULLER SPACES 27
Fix a complexstructure J ∈ J int ( M, ω ) and denote by H J the space of K¨ahler potentials as inRemark 2.5. The analogue of the Mabuchi functional in the present setting is the Ding functional F J : H J → R , defined by(3.26) F J ( h ) := I J ( h ) − log (cid:18)Z M e h ρ J (cid:19) for h ∈ H J , where I J : H J → R is the unique functional that satisfies I J (0) = 0 , d I J ( h ) b h = 1 V Z M b hρ h , ρ h := ω n h n ! , for all h ∈ H J and all b h ∈ Ω ( M ). An explicit formula is I J ( h ) := R
10 1 V R M hρ th dt .For h ∈ H J define θ h := Θ ω h ,J : M → (0 , ∞ ) (see Definition 3.3). Then(3.27) Ric θ h ρ h ,J = ω h , Z M θ h ρ h = 1 , ρ h := ω n h n ! . Since Ric ρ J ,J = ω , we have Ric e h ρ J ,J = ω h = Ric θ h ρ h ,J by (1.3), and hence(3.28) θ h ρ h = e h ρ J R M e h ρ J for all h ∈ H J . This implies(3.29) d F J ( h ) b h = 1 V Z M b hρ h − R M b he h ρ J R M e h ρ J = Z M b h (cid:18) V − θ h (cid:19) ρ h for h ∈ H J and b h ∈ Ω ( M ). Thus the gradient of the Ding functional F J withrespect to the Riemannian metric (2.26) on H J is given by(3.30) grad F J ( h ) = 1 V − θ h for h ∈ H J . In [ ] Berndtsson proved the following. Theorem
Berndtsson) . The Ding functional is convex along geodesics.
Proof.
Let I → H J : t h t be a geodesic so that ∂ t ∂ t h + | d∂ t h | h = 0. Thenit follows from (3.29) that d dt F J ( h ) = 1 V ddt Z M ( ∂ t h ) ρ h − ddt R M ( ∂ t h ) e h ρ J R M e h ρ J = − R M ( ∂ t ∂ t h ) e h ρ J R M e h ρ J − R M ( ∂ t h ) e h ρ J R M e h ρ J + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R M ( ∂ t h ) e h ρ J R M e h ρ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = R M | d∂ t h | h e h ρ J R M e h ρ J − R M ( ∂ t h ) e h ρ J R M e h ρ J + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R M ( ∂ t h ) e h ρ J R M e h ρ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Z M | d∂ t h | h θ h ρ h − Z M ( ∂ t h ) θ h ρ h + (cid:12)(cid:12)(cid:12)(cid:12)Z M ( ∂ t h ) θ h ρ h (cid:12)(cid:12)(cid:12)(cid:12) ≥ . Here the second equality holds because R M ( ∂ t h )( ∂ t ρ h ) = R M | d∂ t h | h ρ h . The lastinequality holds by Lemma 3.7 with ω replaced by ω h , with ρ J replaced by θ h ρ h ,and with F := ∂ t h − R M ( ∂ t h ) θ h ρ h and G := 0. (cid:3) In finite-dimensional GIT the gradient flow of the moment map squared trans-lates into the gradient flow of the Kempf–Ness function. In the present setting themoment map is given by J int ( M, ω ) → Ω ρ ( M ) : J /V − Θ J ), where Θ J isdefined by (3.7). It is convenient to take one eighth (instead of one half) of thesquare of the moment map to obtain the energy functional E ω : J int ( M, ω ) → R defined by(3.31) E ω ( J ) := Z M (cid:16) V − Θ J (cid:17) ω n n !for J ∈ J int ( M, ω ). Consider the Riemannian metric on J int ( M, ω ) determinedby the symplectic form (3.4) and the complex structure b J
7→ − J b J . It is given by h b J , b J i J := Ω DJ ( b J , − J b J ) = Z M (cid:16) trace( b J b J ) − f f − g g (cid:17) ρ J (3.32)for J ∈ J int ( M, ω ) and b J i ∈ T J J int ( M, ω ), where ρ J , f i , g i are as in (3.2) and (3.3).By Lemma 3.4 the differential of the functional E ω in (3.31) is given by d E ω ( J ) b J = Z M f Θ J ρ J = − Ω DJ ( b J, L v J ) = h b J, − J L v J i J , for J ∈ J int ( M, ω ) and b J ∈ T J J int ( M, ω ), where f is as in (3.3), v is the Hamil-tonian vector field of Θ J . and the second equality follows from (3.6). This showsthat the gradient of E ω at J with respect to the metric (3.32) is given by(3.33) grad E ω ( J ) = − J L v J, ι ( v ) ω = d Θ J . Thus a complex structure J ∈ J int ( M, ω ) is a critical point of E ω if and only if theHamiltonian vector field of Θ J is holomorphic. Such a complex structure is called a Donaldson–K¨ahler–Ricci soliton . By (3.33) a negative gradient flow line of E ω is a solution I → J int ( M, ω ) : t J t of the partial differential equation(3.34) ∂ t J t = J t L v t J t , ι ( v t ) ω = d Θ J t . If t J t is a solution of (3.34) on an interval I ⊂ R containing zero with J = J ,and I → Diff ( M ) : t φ t is the isotopy defined by ∂ t φ t + J t v t ◦ φ t = 0, φ = id,then φ ∗ t J t = J for all t and the paths ω t := φ ∗ t ω and θ t := Θ J t ◦ φ t satisfy(3.35) ∂ t ω t = d ( dθ t ◦ J ) , d ( d log( θ t ) ◦ J ) = ω t − Ric ω n t / n ! ,J , Z M θ t ω n t n ! = 1 , This is the
Donaldson–K¨ahler–Ricci flow . Here J is a Fano complex structureand (3.35) is understood as an equation for paths in the space S J in (2.9) ofall symplectic forms that are compatible with J and represent the cohomologyclass 2 πc R ( J ). When ω ∈ S J is fixed, a solution of (3.35) has the form ω t = ω h t ,where I → H J : t h t is a smooth path satisfying(3.36) ∂ t h t = θ h t − V .
By (3.30) the solutions of (3.36) are the negative gradient flow lines of the Dingfunctional F J : H J → R in (3.26). The next remark shows that (3.36) is a secondorder parabolic partial differential equation. Remark . Let ∇ be the Levi-Civita connection of the metric h· , ·i := ω ( · , J · )and let h ∈ H J . Then ρ h = det(1l − ∇ h + J ( ∇ h ) J ) / ω n / n ! and hence it fol-lows from (3.28) that θ h = ( R M e h ρ J ) − e h Θ J det(1l − ∇ h + J ( ∇ h ) J ) − / . ICCI FORM AND TEICHM¨ULLER SPACES 29
Define the functional H ω : J int ( M, ω ) → R by(3.37) H ω ( J ) := Z M log( V Θ J )Θ J ω n n !for J ∈ J int ( M, ω ). This functional was introduced by Weiyong He [ ] (as afunctional on the space of K¨ahler potentials for a fixed complex structure). It isnonnegative and vanishes on a complex structure J ∈ J int ( M, ω ) if and only if itsatisfies the K¨ahler–Einstein condition Ric ω n / n ! ,J = ω (see part (i) of Theorem 3.10below). By Lemma 3.4 the differential of the functional H ω is given by d H ω ( J ) b J = Z M f log(Θ J ) ρ J = h b J, − J L v J i J for J ∈ J int ( M, ω ) and b J ∈ T J J int ( M, ω ), where f is as in (3.3), v is the Hamil-tonian vector field of log(Θ J ), and the second equality follows from (3.6). Thisshows that the gradient of H ω at J with respect to the metric (3.32) is given by(3.38) grad H ω ( J ) = − J L v J, ι ( v ) ω = d log(Θ J ) . Thus a complex structure J ∈ J int ( M, ω ) is a critical point of H ω if and only ifthe Hamiltonian vector field of log(Θ J ) is holomorphic. Such a complex structureis called a K¨ahler–Ricci soliton . By (3.38) a negative gradient flow line of H ω isa solution I → J int ( M, ω ) : t J t of the partial differential equation(3.39) ∂ t J t = J t L v t J t , ι ( v t ) ω = d log(Θ J t ) , If t J t is a solution of (3.39) on an interval I ⊂ R containing zero with J = J ,and I → Diff ( M ) : t φ t is the isotopy defined by ∂ t φ t + J t v t ◦ φ t = 0, φ = id,then φ ∗ t J t = J for all t and the paths ω t := φ ∗ t ω and θ t := Θ J t ◦ φ t satisfy theequation ∂ t ω t = d ( d log( θ t ) ◦ J ). With ρ t := ω n t / n ! we also have θ t ρ t = φ ∗ t ρ J t ,hence Ric ρ t ,J + d ( d log( θ t ) ◦ J ) = Ric θ t ρ t ,J = φ ∗ t Ric ρ Jt ,J t = φ ∗ t ω = ω t , and so(3.40) ∂ t ω t = ω t − Ric ρ t ,J , ρ t := ω n t / n ! . This is the standard
K¨ahler–Ricci flow on the space S J of all J -compatiblesymplectic forms in the class 2 πc R ( J ) associated to a Fano complex structure J .When a symplectic form ω ∈ S J is fixed, a solution of (3.40) has the form ω t = ω h t ,where I → H J : t h t is a smooth path satisfying(3.41) ∂ t h t = log( V θ h t ) . Now define the functional H J : H J → R as in Weiyong He’s original paper [ ] by(3.42) H J ( h ) := H ω h ( J ) = Z M log( V θ h ) θ h ρ h for h ∈ H J , where θ h and ρ h are as in (3.27). The properties of this functional withregard to the K¨ahler–Ricci flow are summarized in the following theorem. The firsttwo assertions are due to He [ ] and the last inequality is due to Donaldson [ ]. Theorem
He, Donaldson) . Fix a complex structure J ∈ J int ( M, ω ) . (i) Let h ∈ H J . Then H J ( h ) ≥ with equality if and only if Ric ρ h ,J = ω h . (ii) A K¨ahler potential h ∈ H J is a critical point of H J if and only if it is a K¨ahler–Ricci soliton, i.e. the ω h -Hamiltonian vector field of log( θ h ) is holomorphic. (iii) Every solution I → H J : t h t of (3.41) satisfies the inequalities (3.43) ddt H J ( h t ) ≤ , ddt F J ( h t ) ≤ −H J ( h t ) . Proof.
Following [ ], we define the function B : (0 , ∞ ) → R by(3.44) B ( x ) := x log( V x ) + 1 V − x for x >
0. Then B ′ ( x ) = log( V x ) and B ′′ ( x ) = 1 /x . Hence B (1 /V ) = 0 and B isstrictly convex. This implies B ( x ) > x = 1 /V and(3.45) (cid:18) x − V (cid:19) log( V x ) = (cid:18) x − V (cid:19) B ′ ( x ) ≥ B ( x ) = x log( V x ) + 1 V − x. To prove part (i), fix an element h ∈ H J . Then, by (3.27), (3.42), and (3.44), H J ( h ) = Z M (cid:18) θ h log( V θ h ) + 1 V − θ h (cid:19) ρ h = Z M B ( θ h ) ρ h . Hence H J ( h ) ≥ θ h = 1 /V . This proves (i).We prove part (ii). A calculation shows that d H J ( h ) b h = − Z M h d b h, d log( V θ h ) i h θ h ρ h + Z M b h log( V θ h ) θ h ρ h − (cid:18)Z M log( V θ h ) θ h ρ h (cid:19) (cid:18)Z M b hθ h ρ h (cid:19) (3.46)for h ∈ H J and b h ∈ Ω ( M ). This implies d H J ( h ) log( V θ h ) = − Z M | d log( V θ h ) | h θ h ρ h + Z M (cid:0) log( V θ h ) (cid:1) θ h ρ h − (cid:18)Z M log( V θ h ) θ h ρ h (cid:19) ≤ h ∈ H J . Here the inequality follows from Lemma 3.7, with ω, ρ J replacedby ω h , θ h ρ h and F := log( V θ h ) − R M log( V θ h ) θ h ρ h and G := 0. It follows also fromLemma 3.7 that d H J ( h ) = 0 if and only if d H J ( h ) log( V θ h ) = 0 if and only if thevector field v defined by ι ( v ) ω h = d log( V θ h ) satisfies L v J = 0. This proves (ii).We prove part (iii). The first inequality in (3.43) follows directly form (3.47).To prove the second inequality, recall from equation (3.29) that d F J ( h ) log( V θ h ) = Z M log( V θ h ) (cid:16) V − θ h (cid:17) ρ h ≤ − Z M (cid:16) θ h log( V θ h ) + 1 V − θ h (cid:17) ρ h = −H J ( h ) . Here the second step follows from (3.45). This proves (iii) and the theorem. (cid:3)
In [ ] Donaldson noted the following. If [0 , ∞ ) → H J : t h t is a solutionof the K¨ahler–Ricci flow (3.41) and the limit h := lim t →∞ h t exists in H J , but thepair ( ω h , J ) is not a K¨ahler–Einstein structure, then it follows from Theorem 3.10that H J ( h t ) ≥ H J ( h ) > F J ( h t ) diverges to minusinfinity as t tends to infinity. This corresponds to the observation in GIT that theKempf–Ness function of an unstable point is unbounded below. The analogue ofthe Kempf–Ness Theorem in the present setting would be the assertion(3.48) inf h ∈ H J Z M (cid:16) V − θ h (cid:17) ρ h > ⇐⇒ inf h ∈ H J F J ( h ) = −∞ for every J ∈ J int ( M, ω ). This seems to be an open question.
ICCI FORM AND TEICHM¨ULLER SPACES 31
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