The coeffective differential complex on a symplectic manifold is extended
both in length and in scope, unifying the constructions of various other
authors.Comment: 9 page

Eastwood, Michael

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The coeffective differential complex on a symplectic manifold is extended both in length and in scope, unifying the constructions of various other authors.The author is supported by the Australian Research Council

The Australian National University

Illinois Journal of MathematicsVolume 57, Number 2, Summer 2013, Pages 373–381S 0019-2082EXTENSIONS OF THE COEFFECTIVE COMPLEXMICHAEL EASTWOODAbstract. The coeffective differential complex on a symplecticmanifold is extended both in length and in scope, unifying theconstructions of various other authors.1. IntroductionThis article is both an addendum to [4] and a precursor to [7]. In [4], wediscussed the construction of differential complexes on manifolds equippedwith various geometric structures. Mostly, these geometries were parabolic [5]but there were two exceptions, specifically contact geometry for which thereis the Rumin complex [14] and symplectic for which there is a very similarcomplex [15], which we dubbed the Rumin–Seshadri complex (it was inde-pendently discovered by Tseng and Yau [17], [18]). This article extends therealm of these complexes, specifically covering conformally symplectic mani-folds and conformally calibrated G2 manifolds (see, for example, [1], [19] and[9], respectively).In [2], T. Bouche introduced a differential complex naturally defined onany symplectic manifold M and coined the term coeffective complex for it(see also [8]). If M has dimension 2n, then it is the subcomplex of the secondhalf of the de Rham complexΛnd−→ Λn+1 d−→ · · · d−→ Λ2n−2 d−→ Λ2n−1 d−→ Λ2n → 0∪ ∪ ∪ ‖ ‖Λn⊥ → Λn+1⊥ → · · · → Λ2n−2⊥ → Λ2n−1 → Λ2n → 0,Received August 9, 2012; received in final form May 30, 2013.The author is supported by the Australian Research Council.2010 Mathematics Subject Classification. Primary 58J10. Secondary 53D05.373c©2014 University of Illinois374 M. EASTWOODwhere, if J denotes the symplectic form, then the bundle Λk⊥ is defined as thekernel of ΛkJ∧−−−→ Λk+2. Under the canonical isomorphismsJ ∧ J ∧ · · · ∧ J︸ ︷︷ ︸n−k∧ : Λk −→ Λ2n−k for k = 0,1,2, . . . , nthe bundle Λ2n−k⊥ may equally well be regarded as a subbundle of Λk, which weshall write as Λk⊥ and, as such, provides a natural complement to the rangeof Λk−2J∧−−−→ Λk for k = 2,3, . . . , n. Using indices (more precisely, abstractindices in the sense of [13]), sections of the bundle Λk⊥ for k = 2,3, . . . , n areprecisely the k-forms that are trace-free with respect to Jab, that is,Jabωabc···d = 0,where Jab is the inverse of Jab (let us say JacJbc = δab, where δab is theKronecker delta). Thus, we may rewrite the coeffective complex as(1) Λn⊥d⊥−−→ Λn−1⊥d⊥−−→ · · · d⊥−−→ Λ2⊥d⊥−−→Λ1 d⊥−−→Λ0 → 0.Bouche [2] showed that it is elliptic except at Λn⊥. Since the diagrams withexact rows0→ Λk−2 J∧−−−→ Λk → Λk⊥ → 0↓ ↓0→ Λk−1 J∧−−−→ Λk+1 → Λk+1⊥ → 0commute, there is a canonically defined differential complex going the otherway:(2) 0→ Λ0 d−→Λ1 d⊥−−→Λ2⊥d⊥−−→ · · · d⊥−−→ Λn−1⊥d⊥−−→ Λn⊥.In fact, one can easily check that (1) and (2) are adjoint to each other underthe pairingΛk⊥ ⊗Λk⊥−→ Λ2n−k⊥ ⊗Λk⊥∧−−−→ Λ2n.The Rumin–Seshadri complex joins (1) and (2) with a symplectically invariantsecond order linear differential operator d(2)⊥ : Λn⊥ → Λn⊥ to obtain an ellipticcomplex(3)0→ Λ0 d−→ Λ1 d⊥−→ Λ2⊥d⊥−→ Λ3⊥d⊥−→ · · · d⊥−→ Λn⊥↓ d(2)⊥0← Λ0 d⊥←− Λ1 d⊥←− Λ2⊥d⊥←− Λ3⊥d⊥←− · · · d⊥←− Λn⊥.In four dimensions this complex is due to Smith [16] and in higher dimensionsit was also found by L.-S. Tseng and S.-T. Yau [17], [18] who go on to studyits cohomology on compact manifolds. The construction of (3) given in [4]will be generalised in the following section.EXTENSIONS OF THE COEFFECTIVE COMPLEX 3752. Conformally symplectic manifoldsA conformally symplectic structure on an even dimensional manifold Mof dimension at least 6 is defined by a non-degenerate 2-form J but, insteadof requiring that J be closed, as one would for a symplectic structure, onerequires only that(4) dJ = 2α∧ Jfor some 1-form α (the factor of 2 being chosen only for convenience). Non-degeneracy of J implies that α is uniquely defined by (4). It is called the Leeform [12]. Differentiating (4) gives0 = d2J = 2dα∧ J + 2α∧ dJ = 2dα∧ J + 4α∧ α∧ J = 2dα∧ Jand, as J ∧ : Λ2 → Λ4 is injective since dimM ≥ 6, we see that α is closed.In dimension 4, equation (4) defines a unique Lee form α and, for the defi-nition of conformally symplectic, we require that α be closed. If we rescaleJ by a positive smooth function, say Ĵ = Ω2J , then (4) remains valid withα replaced by α̂ = α +Υ for Υ ≡ d logΩ. Hence, the notion of conformallysymplectic is invariant under such rescalings (and also in dimension 4 sincedΥ= 0). Locally, we may use this freedom to eliminate α and obtain an ordi-nary symplectic structure. Globally, however, this need not be the case. Forexample, the rescaled symplectic formJ ≡(1/‖x‖)2(dx1 ∧ dx2 + dx3 ∧ dx4 + · · ·)on R2n is invariant under dilation x → λx and, therefore, descends to a confor-mally symplectic structure on S1×S2n−1 whereas there is no global symplecticform on this manifold. If we continue to denote the inverse of Jab by Jab, andconsider the vector field Xa ≡ Jabαb, then the identitiesJadJbeJcf (∇[dJef ]−2α[dJef ]) = Jd[a∇dJbc] − 2X [aJbc],JadJbe(2∇[dαe] + 3Xc(∇[dJec]−2α[dJec]))=−Xc∇cJab − 2Jc[a∇cXb]are readily established for any torsion-free connection ∇a and show that aconformally symplectic structure is equivalent to a Jacobi structure (Jab,Xa)if we insist that Jab be non-degenerate (as discussed in [1]).Theorem 1. On any conformally symplectic manifold (M,J), there is acanonically defined elliptic complex(5)0→ Λ0 → Λ1 → Λ2⊥ → Λ3⊥ → · · · → Λn⊥↓0← Λ0 ← Λ1 ← Λ2⊥ ← Λ3⊥ ← · · · ← Λn⊥,where Λk⊥ denotes the bundle of k-forms that are trace-free with respect to J .All operators are first order except for the middle operator, which is secondorder. In the symplectic case, the second half of the complex coincides with the376 M. EASTWOODcoeffective complex. This complex is locally exact except at Λ0 and Λ1 nearthe beginning.Proof. Consider the diagram(6)−→ Λp d−2α∧−−−−−−→ Λp+1 d−2α∧−−−−−−→ Λp+2 −→↑ J ∧ ↑ J ∧ ↑ J ∧−→ Λp−2 d−−−−→ Λp−1 d−−−−→ Λp −→ .The bottom row is the de Rham complex and, in particular, is locally exactexcept at Λ0. Since dα= 0, the same is true of the top row. Since dJ = 2α∧J ,the diagram commutes. Now consider the columns. In the middle, non-degeneracy of J ensures thatJ ∧ : Λn−1 → Λn+1is an isomorphism. To the left of this, we have injections and, to the right,we have surjections. As discussed in Section 1, the trace-free forms Λk⊥ maybe canonically identified with the cokernel ofJ ∧ : Λk−2 → Λk for k = 2,3, . . . , nbut also with the kernel ofJ ∧ : Λ2n−k →Λ2n−k+2 for k = 2,3, . . . , n.The spectral sequence of a double complex completes the proof. Explicit formulæ for the operators in this complex can be given by usingindices and an arbitrarily chosen torsion-free connection but are quite com-plicated since they necessarily employ the decomposition of arbitrary k-formsinto their trace-free partsΛk =Λk⊥ ⊕Λk−2⊥ ⊕Λk−4⊥ ⊕ · · · for k = 2,3, . . . , ncorresponding to the branching of ΛkR2n under Sp(2n,R) ⊂ SL(2n,R) (cf.the combinatorial formulæ in [18, §2.1]).To discuss the global cohomology of the complex (5), let us relabel its termsas Br for r = 0,1,2, . . . ,2n,2n+ 1 and define(7) HrJ(M)≡ker : Γ(M,Br)→ Γ(M,Br+1)im : Γ(M,Br−1)→ Γ(M,Br) .In comparison with [8] in the symplectic case, we haveHrJ (M) =Hr−1(A(M))for r = n+ 2, n+ 3, . . . ,2n+ 1for their coeffective cohomology but now, for compact M , we have finite-dimensional vector spaces for all r = 0,1,2, . . . ,2n,2n+ 1. Also in the sym-plectic case, these cohomologies were introduced and studied by Tseng andEXTENSIONS OF THE COEFFECTIVE COMPLEX 377Yau [17], [18] and our notation compares as follows.HrJ(M) = PHr∂+(M) for 0≤ r < n,HnJ (M) = PHnddΛ(M), Hn+1J (M) = PHnd+dΛ(M),HrJ(M) = PH2n+1−r∂−(M) for n+ 1< r ≤ 2n+ 1.(Tseng and Yau refer to these and similar cohomologies as ‘primitive.’) Ac-cording to Theorem 1, the cohomology H• of (5) on the level of sheaves ofgerms of smooth functions occurs only at B0 and B1 and, from its proof, wesee that H1 =R. Also H0 is a locally constant sheaf. Specifically,H0 = {f s.t. df − 2fα= 0},and may equivalently be viewed as parallel sections of the trivial bundleequipped with the flat connection defined by −2α as connection form. Inthe symplectic case, we have H0 = R. Evidently, the top row of (6) providesa fine resolution of H0 and so the sheaf cohomology Hr(M,H0) may be iden-tified as the cohomology of the complex Γ(M,Λ•) with ω → dω − 2α ∧ ω asdifferential. The following theorem extends the long exact sequence [8, (5)].Theorem 2. On a conformally symplectic manifold (M,J), we haveH0J(M) =H0(M,H0), H2n+1J (M) =H2n(M,R),and a long exact sequence0→H1(M,H0)→H1J (M)→H0(M,R)δ−→H2(M,H0)→ · · ·→HrJ (M)→Hr−1(M,R)δ−→Hr+1(M,H0)→ · · ·→H2nJ (M)→H2n−1(M,R)→ 0,where δ : Hr−1(M,R)→Hr+1(M,H0) is given by cup product with the coho-mology class [J ] ∈H2(M,H0).Proof. The hypercohomology spectral sequence for the complex B• as acomplex of sheaves reads, at the E2-levelH0(M,H0) H1(M,H0) H2(M,H0) H3(M,H0) · · ·H0(M,R) H1(M,R) H2(M,R) H3(M,R) · · ·~~~~~~~~~~~~~~~~~~~~~~~~~~~~and the desired conclusions follow. (Spectral sequence reasoning can always be replaced by an appropriatediagram chase, in this case on the double complex (6).)As an application of Theorem 2, if we consider complex projective spaceCPn with J its usual Kähler form, then[J ]∪ : Hr−1(CPn,R)→Hr+1(CPn,R)378 M. EASTWOODis an isomorphism for 1≤ r ≤ 2n− 1. Therefore,H0J(CP) =R, HrJ(CPn) = 0 for 1≤ r ≤ 2n, H2n+1J (CPn) =R.More generally, Theorem 2 shows that the cohomology HrJ(M) of a symplecticmanifold is determined by its de Rham cohomology and the action of thesymplectic class [J ] ∈H2(M,R). In particular, there are evident inequalitiesconcerning dimHrJ and the Betti numbers of a compact symplectic manifold(including those of [8, Theorem 3.1]).3. Conformally calibrated G2-manifoldsFollowing [9], a conformally calibrated G2-manifold is defined as a G2-manifold (M,φ) such that(8) dφ= 2α∧ φfor some 1-form α. Recall [3], [6], [9] that φ is the fundamental 3-form defininga reduction of structure group on the 7-dimensional smooth manifold M fromGL(7,R) to G2 ⊂ SO(7)⊂GL(7,R). In parallel with the symplectic case, theform φ may be locally rescaled so that it is closed (and a G2-manifold withclosed fundamental form is said to be calibrated). As in the symplectic caseand as detailed in [9], the form φ, pointwise sometimes known as the Cayleyform [6], is sufficiently non-degenerate thatφ∧ : Λk −→ Λk+3 is injective for k = 0,1,φ∧ : Λ2 −→ Λ5,φ∧ : Λk −→ Λk+3 is surjective for k = 3,4.(9)One way to see this is to decompose the forms on M into G2-irreducibles(10)Λ0 = • •〈0 0 Λ1 = • •〈1 0 Λ2 = • •〈0 1 ⊕ • •〈1 0Λ3 = • •〈2 0 ⊕ • •〈1 0 ⊕ • •〈0 0 Λ4 = • •〈2 0 ⊕ • •〈1 0 ⊕ • •〈0 0Λ5 = • •〈0 1 ⊕ • •〈1 0 Λ6 = • •〈1 0 Λ7 = • •〈0 0and check (9) on the level of highest weights. The canonical Hodge isomor-phism Λk ∼=Λ7−k is evident in this decomposition. Parallel to the symplecticcase, let us writeΛ4⊥ ≡ kerφ∧ : Λ3 → Λ6, Λ3⊥ ≡ kerφ∧ : Λ4 →Λ7and, by inspecting (10), note thatΛ3⊥ = • •〈2 0 ⊕ • •〈1 0 Λ4⊥ = • •〈2 0 ⊕ • •〈0 0also provide canonical complements to the ranges of φ ∧ : Λ0 → Λ3 andφ ∧ : Λ1 → Λ4, respectively. From (9) we see that, as in the conformallysymplectic case, α is uniquely defined by (8) and is closed.EXTENSIONS OF THE COEFFECTIVE COMPLEX 379Theorem 3. On any conformally calibrated G2-manifold (M,φ), there isa canonically defined elliptic complex(11)0→ Λ0 → Λ1 → Λ2 → Λ3⊥ → Λ4⊥↓0← Λ0 ← Λ1 ← Λ2 ← Λ3⊥ ← Λ4⊥.All differential operators are first order except for the middle operator, whichis second order. The second half of this complex coincides with the coeffec-tive complex defined in [8]. It is locally exact except at Λ0 and Λ2 near thebeginning.Proof. Consider the diagram−→ Λp d−2α∧−−−−−−→ Λp+1 d−2α∧−−−−−−→ Λp+2 −→↑ ∧ φ ↑ ∧ φ ↑ ∧ φ−→ Λp−3 d−−−−→ Λp−2 d−−−−→ Λp−1 −→ .The bottom row is the de Rham complex and, in particular, is locally exactexcept at Λ0. Since dα= 0, the same is true of the top row. Since dφ= 2α∧φ,the diagram commutes. The columns behave according to (9). Hence, the firstspectral sequence of this double complex reads, at the E1-level0 0 0 0 0 Λ3⊥ Λ4 Λ5 Λ7Λ0 Λ1 Λ2 Λ3⊥ 0 0 0 0 0→ → →→ → →Passing to the E2-level constructs the complex and the second spectral se-quence identifies its local cohomology H• asH0 = {f s.t. df − 2fα= 0}, H2 =R,with all others vanishing. Finally, ellipticity of this complex is inherited fromthat of the de Rham complex. Specifically, for Λ1  ξ = 0, the symbol complexof (11) is constructed from the double complex−→ Λp ξ∧−−−→ Λp+1 ξ∧−−−→ Λp+2 −→↑ ∧ φ ↑ ∧ φ ↑ ∧ φ−→ Λp−3 ξ∧−−−→ Λp−2 ξ∧−−−→ Λp−1 −→,the rows of which are exact (they are Koszul complexes). As in the (conformally) symplectic case, this construction (and this proofof ellipticity) avoids explicit formulæ for the operators. If such formulæ areneeded, then one simply needs explicitly to write out the branching (9) (as isdone in [9, p. 365]).380 M. EASTWOODAs in the conformally symplectic case (7), we may consider the globalcohomology on M of the complex (11), which we shall denote by Hrφ(M) for0≤ r ≤ 9.Theorem 4. On a conformally calibrated G2-manifold (M,φ), we haveH0φ(M) =H0(M,H0), H1φ(M) =H1(M,H0),H8φ(M) =H6(M,R), H9φ(M) =H7(M,R)and a long exact sequence0→H2(M,H0)→H2φ(M)→H0(M,R)δ−→H3(M,H0)→H3φ(M)→H1(M,R)δ−→H4(M,H0)→ · · ·· · · →H6φ(M)→H4(M,R)δ−→H7(M,H0)→H7φ(M)→H5(M,R)→ 0,where δ : Hr(M,R) → Hr+3(M,H0) is given by cup product with the coho-mology class [φ] ∈H3(M,H0).Proof. Immediate from the hypercohomology spectral sequence as for theproof of Theorem 2 except that the connecting homomorphism δ does notappear until the E3-level. In the calibrated case (when α = 0), Hr(M,H0) =Hr(M,R) and we seethat Hrφ(M) is determined by the de Rham cohomology of M and the actionof [φ] ∈H3(M,R) by cup product.4. Other geometriesThere are several other geometries defined by special k-forms for whichone can apply similar reasoning. Certainly, there are Spin(7)-geometries indimension 8 defined [3] by a fundamental 4-form Φ. The construction givenin this article extends to this case and, by the work of Joyce [11], there arenon-trivial compact examples with dΦ= 0.Also, there are SO(3)× SO(3)-geometries in dimension 9 defined [10] by afundamental 5-form and SU(4)×U(1)-geometries in dimension 10 defined [6]by a fundamental 6-form or 4-form but, for the moment, it is unclear whetherthere are any examples of such geometries that are not locally homogeneous.References[1] A. Banyaga, Symplectic geometry and related structures, Cubo 6 (2004), 123–138. MR 2077147[2] T. Bouche, La cohomologie coeffective d’une variété symplectique, Bull. Sci. Math.114 (1990), 115–122. MR 1056157[3] R. L. Bryant, Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987), 525–576. MR 0916718EXTENSIONS OF THE COEFFECTIVE COMPLEX 381[4] R. L. Bryant, M. G. Eastwood, A. R. Gover and K. 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MR 0809073Michael Eastwood, Mathematical Sciences Institute, Australian National Uni-versity, ACT 0200, AustraliaE-mail address: meastwoo@member.ams.org

Extensions of the coeffective complex

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