aa r X i v : . [ m a t h . DG ] A ug ALMOST HERMITIAN IDENTITIES
JOANA CIRICI AND SCOTT O. WILSON
Abstract.
We study the local commutation relation between the Lefschetzoperator and the exterior differential on an almost complex manifold with acompatible metric. The identity that we obtain generalizes the backbone of thelocal K¨ahler identities to the setting of almost Hermitian manifolds, allowingfor new global results for such manifolds. Introduction
On a K¨ahler manifold (
M, J, ω ), the most fundamental local identity is perhapsthe commutation relation between the exterior differential d and the adjoint Λ tothe Lefschetz operator,(1) [Λ , d ] = ⋆ I − d I ⋆, where ⋆ denotes the Hodge star operator and I denotes the extension of J to allforms.This identity, due to A. Weil [Wei58], strongly depends on the K¨ahler condition, dω = 0, and in fact is true when removing the integrability condition N J ≡
0. So,it is valid for almost K¨ahler and also symplectic manifolds as well [dBT01, TT20,CW20]. On the other hand, there is also a generalization of the K¨ahler identitiesin the Hermitian setting (see [Dem86, Wil20]), which strongly uses integrability.When the manifold is only almost Hermitian, then the above local identity doesnot hold in general, as noticed implicitly in [Ohs82]. The purpose of this short noteis to show precisely how the above K¨ahler identity (1) becomes modified when theform ω is not closed.The main result is given in Theorem 3.1 below, which has several applicationsincluding the uniqueness of Dirichlet problem ∂ ¯ ∂u = g with u | ∂ Ω = φ, on any compact domain Ω in an almost complex manifold. This in turn implies thatthe Dolbeault cohomolgy introduced in [CW18], for all almost complex manifolds,satisfies H , ( M ) ∼ = C for a compact connected almost complex manifold.Another application of the almost Hermitian identities of Theorem 3.1 appearsin forthcoming work by Feehan and Leness [FL]. There the fundamental relation ofProposition 4.1 is used to show that the moduli spaces of unitary anti-self-dual con-nections over any almost Hermitian 4-manifold is almost Hermitian, whenever theNijenhuis tensor has sufficiently small C -norm. This generalizes a well known re-sult for K¨ahler manifolds that was exploited in Donaldson’s work in the 1980’s, and J. Cirici would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016-76453-C2-2-P) and the Serra H´unter Program.S. Wilson acknowledges support provided by a PSC-CUNY Award, jointly funded by TheProfessional Staff Congress and The City University of New York. is expected to have consequences for the topology of almost complex 4-manifoldswhich are of so-called Seiberg-Witten simple type.When M is compact, local identities lead to consequences in cohomology, oftengoverned by geometric-topological inequalities. Indeed, the exterior differentialinherits a bidegree decomposition into four components d = ¯ µ + ∂ + ∂ + µ and theHermitian metric allows one to consider the Laplacian operators associated to eachof these components. In the compact case, the numbers ℓ p,q := dim Ker (∆ ¯ ∂ + ∆ µ ) | ( p,q ) given by the kernel of ∆ ¯ ∂ +∆ µ in bidegree ( p, q ) are finite by elliptic operator theory.When J is integrable (and so M is a complex manifold) the operator ∆ µ vanishesand these are just the Hodge numbers ℓ p,q = h p,q . In this case, the Hodge-to-deRham spectral sequence gives inequalities X p + q = k ℓ p,q ≥ b k , where b k denotes the k -th Betti number. On the other hand, as shown in [CW20],one main consequence of the local identity (1) in the almost K¨ahler case dω = 0 isthe converse inequality X p + q = k ℓ p,q ≤ b k . Of course, in the integrable K¨ahler case both inequalities are true and so one re-covers the well-known consequence of the Hogde decomposition X p + q = k ℓ p,q = b k . The local identities of [Dem86, Wil20] for complex non-K¨ahler manifolds includeother algebra terms which lead to further Laplacian operators, leading also to var-ious inequalities relating the geometry with the topology of the manifold.With this note, we aim to further understand the origin of these inequalities bymeans of the correct version of (1) for almost Hermitian manifolds for which, apriori, the only geometric-topological inequality in the compact case is given by X p + q = k dim Ker (∆ ¯ µ + ∆ ¯ ∂ + ∆ ∂ + ∆ µ ) | ( p,q ) ≤ b k . Acknowledgments.
The authors would like to thank Paul Feehan for encouragingus develop some previous notes into the present paper.2.
Preliminaries
Let ( A , d ) denote the complex valued differential forms of an almost complexmanifold ( M, J ). For any Hermitian metric, define the associated Hodge-star op-erator ⋆ : A p,qx → A n − q,n − px by ω ∧ ⋆ ¯ η = h ω, η i vol , where ω is the fundamental (1 , n ! ω n ∈ A n,n is the volume formdetermined by the Hermitian metric. Note ⋆ = ( − k on A k .Define d ∗ = − ⋆ d⋆ , so that d ∗ ⋆ = ( − k +1 ⋆ d on A k . Similarly, consider thebidegree decomposition of the exterior differential d = ¯ µ + ¯ ∂ + ∂ + µ, LMOST HERMITIAN IDENTITIES 3 where the bidegree of each component is given by | ¯ µ | = ( − , , | ¯ ∂ | = (0 , , | ∂ | = (1 ,
0) and | µ | = (2 , − . We then let ¯ δ ∗ = − ⋆ δ⋆ for δ = ¯ µ, ¯ ∂, ∂, µ and we have the bidegree decomposition d ∗ = ¯ µ ∗ + ¯ ∂ ∗ + ∂ ∗ + µ ∗ . where | ¯ µ ∗ | = (1 , − , | ¯ ∂ ∗ | = (0 , − , | ∂ ∗ | = ( − ,
0) and | µ ∗ | = ( − , . Let L : A p,q → A p +1 ,q +1 be the real (1 , L ( η ) = ω ∧ η . LetΛ = L ∗ = ⋆ − L⋆ . Then ⋆ Λ = L⋆ and ⋆L = Λ ⋆ . Let P k = Ker Λ ∩ A k denote theprimitive forms of total degree k .It is well known that { L, Λ , [ L, Λ] } defines a representation of sl (2 , C ) and inducesthe Lefschetz decomposition on forms: Lemma 2.1.
We have A k = k/ M r =0 L r P k − r , and this direct sum decomposition respects the ( p, q ) bigrading. Let [
A, B ] = AB − ( − | A || B | BA be the graded commutator, where | A | denotesthe total degree of A . This defines a graded Poisson algebra[ A, BC ] = [
A, B ] C + ( − | A || B | B [ A, C ]The following is well known (e.g. [Huy05] Corollary 1.2.28):
Lemma 2.2.
For all j ≥ and α ∈ A k [ L j , Λ] α = j ( k − n + j − L j − α. By induction, and the fact that [ d, L ] and L commute, we have: Lemma 2.3.
For all n ≥ d, L n ] = n [ d, L ] L n − , and ⋆ [ d, L ] α = ( − k +1 [ d ∗ , Λ] ⋆ α for α ∈ A k . Let I be the extension of J to all forms as an algebra map with respect to wedgeproduct, so that I p,q acts on A p,q by multiplication by i p − q . Then I p,q = ( − p + q so that I − p,q = ( − p + q I p,q . Note that I and ⋆ commute, and I and L n commute forall n ≥
0. The following is a direct calculation.
Lemma 2.4.
If an operator T r,s : A p,q → A p + r,q + s has bidegree ( r, s ) , then I − r + p,s + q ◦ T r,s ◦ I p,q = ( − i ) r − s T r,s . The above result readily implies that I − ◦ d ◦ I = − i (¯ µ − ¯ ∂ + ∂ − µ ) . Finally, the following is well known (e.g. [Huy05] Proposition 1.2.31):
Lemma 2.5. If M is an almost Hermitian manifold of dimension n , then for all j ≥ and all α ∈ P k , ⋆L j α = ( − k ( k +1)2 j !( n − k − j )! L n − k − j I α. J. CIRICI AND S. WILSON Almost Hermitian identities
By the previous section, any differential form η can be written as η = L j α forunique j, k ≥ α ∈ P k . We now state the main result: Theorem 3.1.
For any almost Hermitian manifold of dimension n , let α ∈ P k ,with dα written as (2) dα = α + Lα + L α + · · · , for unique α r ∈ P k +1 − r . Then, for all j ≥ , [Λ , d ] L j α − ⋆ I − d I ⋆ L j α = 1 j + 1 I − [ d ∗ , Λ] I L j +1 α + j Λ[ d, L ] L j − α + j ( j − k − n + j − d, L ] L j − α + ∞ X r =2 f n,j,k ( r ) L j + r − α r , where f n,k,j ( r ) = ( r ( n − k + r ) − j ) + ( − r j !( n − k − j + r )!( j + r − n − k − j )! . Remark 3.2.
In the almost K¨ahler case we have [ d ∗ , Λ] = [ d, L ] = 0, and dα = α + Lα , so we recover the identity[Λ , d ] = ⋆ I − d I ⋆, as expected. Proof.
The proof consists of several calculations using the lemmas in the previoussection. Using [ I , L ] = 0, and I = ( − k on A k , we have ⋆ I − d I ⋆ η = ⋆ I − d I (cid:18) ( − k ( k +1)2 j !( n − k − j )! L n − k − j I α (cid:19) = ( − k ( k +1)2 + k j !( n − k − j )! ⋆ I − dL n − k − j α. By Lemma 2.3 this is equal to(3)( − k ( k +1)2 + k j !( n − k − j )! ⋆ I − L n − k − j dα +( − k ( k +1)2 + k j !( n − k − j − ⋆ I − [ d, L ] L n − k − j − α. We first simplify each of these last two summands. By Equation (2), the fact that ⋆ commutes with I , and Lemma 2.5 applied to α r ∈ P k +1 − r , the first summandof Equation (3) is equal to:( − k ( k +1)2 + k j !( n − k − j )! ⋆ I − ∞ X r =0 L n − k − j + r α r ! = ( − k ( k +1)2 + k j !( n − k − j )! I − ∞ X r =0 ( − ( k +1 − r )( k − r +2)2 ( n − k − j + r )!( j + r − L j + r − I α r ! = ∞ X r =0 ( − r +1 j !( n − k − j + r )!( j + r − n − k − j )! L j + r − α r . LMOST HERMITIAN IDENTITIES 5
For the second summand, we use the fact that for all m ≥ β ∈ A k , ⋆L m [ d, L ] β = ⋆ [ d, L ] L m β = ( − k +1 [ d ∗ , Λ] ⋆ L m β. So, the second summand in Equation (3) is equal to( − k ( k +1)2 + k j !( n − k − j − ⋆ I − [ d, L ] L n − k − j − α = ( − k ( k +1)2 +1 j !( n − k − j − I − [ d ∗ , Λ] ⋆ L n − k − j − α = ( − k ( k +1)2 +1 j !( n − k − j − I − [ d ∗ , Λ]( − k ( k +1)2 ( n − k − j − j + 1)! L j +1 I α = − j + 1 I − [ d ∗ , Λ] I L j +1 α, where in the second to last step we used Lemma 2.5.In summary, we have(4) ⋆ I − d I ⋆η = ∞ X r =0 ( − r +1 j !( n − k − j + r )!( j + r − n − k − j )! L j + r − α r − j + 1 I − [ d ∗ , Λ] I L j +1 α. We now compute [Λ , d ] η , by first computing Λ dL j α , using that all α r are prim-itive. By Equation(2), Lemma 2.2, and Lemma 2.3, we have:Λ dL j α = Λ L j dα + Λ[ d, L j ] α = Λ L j ∞ X r =0 L r α r ! + j Λ[ d, L ] L j − α = ∞ X r =0 Λ L j + r α r + j Λ[ d, L ] L j − α = − ∞ X r =0 ( j + r )( k + 1 − r − n + j + r − L j + r − α r + j Λ[ d, L ] L j − α. Next using, α is primitive, and Lemma 2.2 again, we have d Λ L j α = − j ( k − n + j − dL j − α = − j ( k − n + j − L j − dα − j ( k − n + j − j − d, L ] L j − α = − j ( k − n + j − ∞ X r =0 L j + r − α r ! − j ( k − n + j − j − d, L ] L j − α. So,[Λ , d ] η = ∞ X r =0 ( r ( n − k + r ) − j ) L j + r − α r + j Λ[ d, L ] L j − α + j ( j − k − n + j − d, L ] L j − α. J. CIRICI AND S. WILSON
Using this last equation and combining with Equation (4) we obtain the desiredresult:[Λ , d ] η − ⋆ I − d I ⋆ η = 1 j + 1 I − [ d ∗ , Λ] I L j +1 α + j Λ[ d, L ] L j − α + j ( j − k − n + j − d, L ] L j − α + ∞ X r =0 f n,k,j ( r ) L j + r − α r , where f n,k,j ( r ) = ( r ( n − k + r ) − j ) + ( − r j !( n − k − j + r )!( j + r − n − k − j )! . It is a curious fact that f (0) = f (1) = 0, whereas for r ≥ f ( r ) is in generalnon-zero. (cid:3) Applications
On an almost K¨ahler manifold, using the bidegree decompositions of d and d ∗ ,one may derive from (1) the relation[Λ , ∂ ] = i ¯ ∂ ∗ , involving Λ, ∂ and the adjoint of ¯ ∂ . For a non-K¨ahler Hermitian manifold there isan additional term [Λ , ∂ ] = i ( ¯ ∂ ∗ + ¯ τ ∗ )where ¯ τ = [Λ , [ ¯ ∂, L ]] is the zero-order torsion operator (see [Dem86, Wil20]). In thecase of (0 , q )-forms this givesΛ ∂α = i ¯ ∂ ∗ α + i [Λ , ¯ ∂ ∗ ] Lα.
Next we use Theorem 3.1 to derive this local identity also in the non-integrablecase.
Proposition 4.1.
For all α ∈ A ,q in an almost Hermitian manifold we have Λ ∂α = i ¯ ∂ ∗ α + i [Λ , ¯ ∂ ∗ ] Lα.
Proof.
By bidegree reasons α is a primitive form and we have dα = α + Lα + L α where α i are primitive. By expanding each term in the equality of Theorem 3.1with respect to the bidegree decomposition d = ¯ µ + ¯ ∂ + ∂ + µ , in the case j = 0,we obtain: [Λ , d ] α = Λ dα = Λ( ∂ + µ ) α,⋆ I − d I ⋆ α = i ( ¯ ∂ ∗ − ¯ µ ∗ ) α, and I − [ d ∗ , Λ] I Lα = i [Λ , ¯ ∂ ∗ − ¯ µ ∗ ] Lα.
In particular, all terms decompose into sums of pure bidegrees (0 , q −
1) and (1 , q − f n, ,k (2) Lα given in Theorem 3.1 has pure bidegree (1 , q − α must have bidegree(0 , q − , q −
1) we obtain the desiredidentity. (cid:3)
LMOST HERMITIAN IDENTITIES 7
Remark 4.2.
The proof of Proposition 4.1 gives a second identity relating theoperators Λ, µ and ¯ µ and their adjoints, which also contains the term f n, ,k (2) Lα .For forms in A , , this extra term vanishes by bidegree reasons, since α = 0. Thenthe second identity reads Λ µα = − i ¯ µ ∗ α − i [Λ , ¯ µ ∗ ] Lα.
This corrects the identity [Λ , µ ] = − i ¯ µ ∗ known in the almost K¨ahler case for arbitrary forms (see [CW20]).The previous proposition can be used to give a uniqueness result for the Dirichletproblem on compact domains with boundary. Corollary 4.3.
Let Ω be a compact domain in an almost complex manifold ( M, J ) ,with smooth boundary, and let g : Ω → C , and φ : ∂ Ω → C be smooth. Then theDirichlet problem, ∂ ¯ ∂u = g with u | ∂ Ω = φ, has at most one solution u : Ω → C .In particular, if ( M, J ) is a compact connected almost complex manifold, and f : M → C is a smooth map of almost complex manifolds, then f is constant.Proof. It suffices to show the only solution to the homogenous equation with g = 0is a constant function.In any coordinate chart ψ : V → R n containing any maximum point, we pull-back J to ψ ( V ) and consider the J -preserving map u ◦ ψ − : ψ ( V ) → C . Thecomponents of d are natural with respect to this J -preserving map and we use thecompatible metric on ψ ( V ) to define Λ and ¯ ∂ ∗ . Then by Proposition 4.1 with q = 1we obtain − i Λ ∂ ¯ ∂u = ¯ ∂ ∗ ¯ ∂u + [Λ , ¯ ∂ ∗ ] L ¯ ∂u on ψ ( V ). Note ¯ ∂ ∗ ¯ ∂ is quadratic, self-adjoint, and positive, and [Λ , ¯ ∂ ∗ ] L ¯ ∂ is firstorder since [Λ , ¯ ∂ ∗ ] = [ d, L ] ∗ is zeroth order, because [ d, L ] η = dω ∧ η . Then theright hand side is zero, so the maximum principle due to E. Hopf applies [Hop02],showing u is constant in a neighborhood of the maximum point and therefore, byconnectedness, u is constant.The final claim follows taking Ω = M , with empty boundary, g = 0, and notingthe condition that f is a map of almost complex manifolds implies ¯ ∂f = 0. (cid:3) Remark 4.4.
In [CW18], we introduce a Dolbeault cohomology theory that isvalid for all almost complex manifolds. The above corollary is key in showing that,for a compact connected almost complex manifold, this cohomology is well-behavedin lowest bidegree, in the sense that H , ( M ) ∼ = C .Finally, we refer the reader to the work of Feehan and Leness [FL], where therelation of Proposition 4.1, for q = 1, is used to show that the moduli spaces ofunitary anti-self-dual connections over any almost Hermitian 4-manifold is almostHermitian, whenever the Nijenhuis tensor has sufficiently small C -norm. J. CIRICI AND S. WILSON
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Proc. Amer. Math. Soc. ,148(7):3039–3045, 2020.(J. Cirici)
Department of Mathematics and Computer Science, Universitat de Barcelona,Gran Via 585, 08007 Barcelona
E-mail address : [email protected] (S. Wilson) Department of Mathematics, Queens College, City University of NewYork, 65-30 Kissena Blvd., Flushing, NY 11367
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