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A Moment Map Interpretation of the Ricci Form, K\" Ahler--Einstein Structures, and Teichm\" Uller Spaces

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A Moment Map Interpretation of the Ricci Form, K\ 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 moves on to moment map interpretations of the K¨ahler–Einstein condition and the scalar curvature (Quillen–Fujiki–Donaldson). The second section examines the ramifications of these results for various Teichm¨uller spaces and their Weil– Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form 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 the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. A direct consequence of this observation is the Quillen– Fujiki–Donaldson Theorem about the scalar curvature as a moment map for the action of the group of Hamiltonian symplectomorphisms on the space of compatible almost complex structures on a symplectic manifold. This section also discusses how the K¨ahler–Einstein condition can be interpreted as a moment map equation. 1.1. The Ricci form as a moment map. Let M be a closed oriented 2n- manifold equipped with a positive volume form ρ ∈ Ω2n(M). Then the space J (M) arXiv:2004.08659v1 [math.SG] 18 Apr 2020 of all almost complex structures on M that are compatible with the orientation can be thought of as an infinite-dimensional symplectic manifold. Its tangent space at J ∈ J (M) is the space of all complex anti-linear endomorphisms J : TM → TM of the tangent bundle (see [32, Section 2]) and thus can be identified with the 0,1 space ΩJ (M,TM) of complex anti-linear 1-forms on M with valuesb in the tangent bundle. The symplectic form Ωρ is given by 1 (1.1) Ωρ,J (J1, J2) := 2 trace J1JJ2 ρ ZM 0,1 for J ∈ J (M) and J1, J2 ∈ TbJ Jb (M)=ΩJ (M,TMb ).b 1991 Mathematics Subjectb b Classification. 53D20, 53Q20, 53Q25, 14J10. Key words and phrases. moment map, Ricci form, K¨ahler–Einstein, Teichm¨uller space. 1 2 OSCAR GARC´IA-PRADA AND DIETMAR SALAMON The symplectic form is preserved by the action of the group Diff(M,ρ) of volume preserving diffeomorphisms. Denote the identity component by Diff 0(M,ρ) and the subgroup of exact volume preserving diffeomorphisms (that are isotopic to the 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 J0(M) ⊂ J (M) of all almost complex structures that are compatible with the orientation and have real first Chern class zero. It was ex shown in [32] that the action of Diff (M,ρ) on J0(M) is Hamiltonian and that twice the Ricci form appears as a moment map. To make this precise, note that the Lie algebra of Diff ex(M,ρ) is the space of exact divergence-free vector fields and can be identified with the quotient space Ω2n−2(M)/ ker d via the correspon- 2n−2 ex dence Ω (M) → Vect (M,ρ): α 7→ Yα, defined by ι(Yα)ρ = dα. The dual space of the quotient Ω2n−2(M)/ ker d can formally be thought of as the space of exact 2-forms, in that every exact 2-form τ ∈ dΩ1(M) gives rise to a 2n−2 R continuous linear functional Ω (M)/ ker d → :[α] 7→ M τ ∧ α. The Ricci form Ric ∈ Ω2(M) associated to a volume form ρ and an almost ρ,J R complex structure J, both inducing the same orientation of M, is defined by 1 ∇ 1 1 ∇ (1.2) Ricρ,J (u, v) := 2 trace JR (u, v) + 4 trace (∇uJ)J(∇v J) + 2 dλJ (u, v) for u, v ∈ Vect(M). Here ∇ is a torsion-free connection on TM that preserves the ∇ 1 volume form ρ and the 1-form λJ ∈ Ω (M) is defined by ∇ λJ (u) := trace (∇J)u 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 cohomology R class 2πc1 (J). Its dependence on the volume form is governed by the identity 1 (1.3) Ricef ρ,J = Ricρ,J + 2 d(df ◦ J). 0 for J ∈ J (M) and f ∈ Ω (M), and the map (ρ,J) 7→ Ricρ,J is equivariant under the action of the diffeomorphism group, i.e. ∗ (1.4) Ricφ∗ρ,φ∗J = φ Ricρ,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 [23] for sections of certain SL(2n, R) fiber bundles. If ρ is the volume form of a K¨ahler metric and ∇ is the Levi-Civita connection, then ∇J =0 and hence the last two terms in (1.2) vanish and Ricρ,J is the standard Ricci form. In general, the second summand in (1.2) is a correction term which gives rise to a closed 2-form that represents 2π times the first Chern class, and the last summand is a further correction term that makes the Ricci form independent of the choice of the 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 λJ = 0. In the integrable case the 2-form iRicρ,J is the curvature of the Chern connection on the canonical bundle associated to the Hermitian structure determined by ρ and hence is a (1, 1)-form. RICCI FORM AND TEICHMULLER¨ SPACES 3 ex Theorem 1.1 ([32]). The action of the group Diff (M,ρ) on the space J0(M) with the symplectic form (1.1) is a Hamiltonian group action and is generated by 1 the Diff(M,ρ)-equivariant moment map J0(M) → dΩ (M): J 7→ 2Ricρ,J , i.e. (1.5) 2Ricρ(J, J) ∧ α =Ωρ,J (J, LYα J) ZM d0,1 b b 2n−2 for all J ∈ J0(M), all J ∈ ΩJ (M,TM) and all α ∈ Ω (M), where d b Ricρ(J, J) := dt t=0 Ricρ,Jt for any smooth path R satisfying and d . → Jd(M): tb7→ Jt J0 = J dt t=0 Jt = J Theorem 1.1 is based on ideas in [23]. We emphasize that equation (1.5)b does not require the vanishing of the first Chern class. Its proof in [32] relies on the construction of a 1-form Λρ on J (M) with values in the space of 1-forms on M. 0,1 1 For J ∈ J (M) and J ∈ ΩJ (M,TM) the 1-form Λρ(J, J) ∈ Ω (M) is defined by 1 (1.6) bΛρ(J, J) (u) := trace (∇J)u + 2 JJb∇uJ for u ∈ Vect(M), where ∇ bis a torsion-free ρ-connection.b b As before, Λρ(J, J) is independent of the choice of ∇. Moreover, Λρ satisfies the following identities. b Proposition . 0,1 1.2 ([32]) Let J ∈ J (M), J ∈ ΩJ (M,TM), and v ∈ Vect(M). Denote the divergence of v by fv := dι(v)ρ/ρ. Then b (1.7) d Λρ(J, J) = 2Ricρ(J, J), 1 (1.8) Λρ(J, J) ∧ ι(vb)ρ = 2d traceb JJLvJ ρ, ZM ZM (1.9) Λbρ(J, LvJ)=2ι(v)Ricρ,J −b dfv ◦ J + dfJv. For a proof of Proposition 1.2 see [32, Theorems 2.6 & 2.7], and note that equation (1.5) in Theorem 1.1 follows directly from (1.7) and (1.8) with v = Yα. Remark 1.3. Two useful equations (see [32, Lemma 2.12]) are ¯ ¯∗ ∗ (1.10) LX J =2J∂J X, Λρ(J, J)= ι(2J∂J J )ω 0,1 for J ∈ J (M), J ∈ ΩJ (M,TM), X ∈ Vect(Mb). Here ω isb a nondegenerate 2-form on M such that ωn/n!= ρ and h·, ·i := ω(·,J·) is a Riemannian metric. b For any Hamiltonian group action the zero set of the moment map is invariant under 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 isotopy classes of Ricci-flat almost complex structures given by ex W0(M,ρ) := J0(M,ρ)/Diff (M,ρ), (1.11) J0(M,ρ) := {J ∈ J0(M) | Ricρ,J =0} . In the finite-dimensional setting it follows directly from the definitions that an element 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. It was shown in [32, Theorem 2.11] that this carries over to the present situation. 4 OSCAR GARC´IA-PRADA AND DIETMAR SALAMON Proposition 1.4 ([32]). Fix an element J ∈ J (M). 1 0,1 (i) Let λ ∈ Ω (M). Then there exists a J ∈ ΩJ (M,TM) such that Ricρ(J, J)= dλ 2n−2 if and only if M dλ ∧ α =0 for all α ∈ Ω (M) with LYα J =0.
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