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Extremal almost-K\"ahler metrics

We generalize the notions of the Futaki invariant and extremal vector field of a compact K\"ahler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-K\"ahler metrics saturating LeBrun's estimates.


Introduction
Let (M 2n , ω) be a compact symplectic manifold of dimension 2n. Recall that an almost-complex structure J is compatible with ω if the tensor field g(·, ·) = ω(·, J·) defines a Riemannian metric on M ; in this case, the triple (ω, J, g) is referred to as an (ω-compatible) almost-Kähler structure on M . Any such structure defines, in a canonical way, a hermitian connection ∇ on the complex tangent bundle (T (M ), J, g). Taking trace and contracting the curvature of ∇ by ω, one obtains the hermitian scalar curvature s ∇ of (ω, J, g).
It is well-known [8,12] that the space of all ω-compatible almost-Kähler structures, here denoted by AK ω , is a contractible Fréchet manifold endowed with a formal Kähler structure. The infinite dimensional group Ham(M, ω) of hamiltonian symplectomorphisms naturally acts on AK ω and a crucial observation of Donaldson [8] (generalizing [12] to the non-integrable almost-Kähler case) is that this action is hamiltonian with moment map µ : AK ω → (Lie(Ham(M, ω)) * given by µ J (f ) = M s ∇ f ω n n! , where f is any smooth function with zero integral on M , viewed also as an element of the Lie algebra of Ham(M, ω). As already observed in [5], this interpretation of s ∇ immediately implies that the critical points of the functional J → M s ∇ 2 ω n n! over AK ω are the almost-Kähler metrics for which the symplectic gradient of the hermitian scalar curvature is an infinitesimal isometry of the almost-complex structure J. Motivated by striking analogy with the notion of extremal Kähler metric introduced by Calabi [7], we refer to the almost-Kähler metrics verifying the above condition as extremal almost-Kähler metrics.
The above formal picture, restricted to the subspace of diffeomorphic integrable ω-compatible almost-Kähler structures, gives many insights in the theory of extremal Kähler metrics, where the leading conjectures are derived by a considerable scope of analogy with the well-established GIT in the finite dimensional case [11]. It also suggests that extremal almost-Kähler metrics would provide a natural extension of the theory of extremal Kähler metrics to the non-integrable case. In fact, this link has already become explicit in the toric case [9], where the existence of an extremal Kähler metric is conjecturally equivalent to the existence of (infinitely many) non-integrable extremal almost-Kähler metrics; this link was also used in [4] to find an explicit criterion to test K-stability (and therefore (non) existence of extremal Kähler metrics) on projective plane bundles over a curve and construct explicit examples of extremal almost-Kähler metrics. Besides the now appealing motivation, a systematic study of extremal almost-Kähler metric is still to come (see however [5,18,21]).
In this paper, we generalize the notion of the Futaki invariant and extremal vector field to the general almost-Kähler case. This amounts to the observation that fixing a compact subgroup G ⊂ Ham(M, ω) and considering G-invariant ω-compatible almost-Kähler structures (g, J), the L 2 -projection of the hermitian scalar curvature to the finite dimensional space of hamiltonians of elements of Lie(G) is independent of (g, J). This fact, which easily follows from the formal picture described above and which is certainly known to experts (see e.g. [4,16]), is established by a direct argument in Sec. 3.2 below. By taking G be a maximal torus T ⊂ Ham(M, ω), we show in Lemma 3.6 that the projection of the hermitian scalar curvature of any T -invariant compatible almost-Kähler metric defines a Killing potential, z T ω , which must coincide with the hermitian scalar curvature of any T -invariant extremal almost-Kähler metric (should it exist). We call the vector field Z T ω = grad ω z T ω the extremal vector field relative to T. Its vanishing is an obstruction to the existence of T -invariant almost-Kähler metrics of constant hermitian scalar curvature. Noting that Z T ω doesn't change under a T -invariant isotopy of ω, it naturally generalizes the extremal vector field introduced by [14] in the Kähler context.
The main technical input of the paper is proving that Z T ω has closed orbits when [ ω 2π ] is an integral class modulo torsion. This extends the corresponding result in the Kähler case [25], but we need to adapt the 'symplectic' approach of [15] which relies on the localization formula [10]. Yet, the main technical ingredient is our Lemma 2.6 which essentially computes the momentum map of the T -action, with respect to the hermitian Ricci form ρ ∇ .
The structure of the paper is as follows. In Sec. 2, we introduce the necessary background of almost-Kähler geometry with special attention to holomorphic vector fields on almost-Kähler manifolds. In particular, we obtain Bochner formulae involving the hermitian Ricci form ρ ∇ and the so-called * -Ricci form ρ * and derive some vanishing results (Corollaries 2.5 and 2.7). We define in Sec. 3 the extremal vector field relative to a maximal torus T ⊂ Ham(M, ω) in the general almost-Kähler case and prove its periodicity when ω 2π is integral modulo torsion. In Sec. 4, we give an explicit (local) formula of the hermitian scalar curvature in Darboux coordinates which allows us to recast the expressions of the hermitian Ricci form and the scalar curvature in the toric case [1,9]. We then specialise to the 4-dimensional case and construct infinite dimensional families of non-integrable extremal almost-Kähler metrics by using local toric symmetry; this allows us to obtain examples of non-integrable Hermitian-Einstein almost-Kähler metrics saturating LeBrun's estimates [20]. Finally, in Sec. 5, we discuss some avenues of further research.

Preliminaries
2.1. Almost-Kähler structures. An almost-Kähler structure on a real manifold M of dimension 2n is given by a triple (ω, J, g) of a symplectic form ω, an almostcomplex structure J and a Riemannian metric g, which satisfy the compatibility relation (2.1) g(·, ·) = ω(·, J·).
We say that the almost-complex structure J is ω-compatible if it induces a Riemannian metric via (2.1). If, additionally, the almost complex structure J is integrable, then we have a Kähler structure on M . By the Newlander-Nirenberg theorem, the almost-complex structure J is integrable if and only if the Nijenhuis tensor N vanishes; here [·, ·] stands for the Lie bracket.
In particular, the bundle of 2-forms decomposes under the action of J is the subbundle of the primitive J-invariant 2-forms (i.e. 2-forms pointwise orthogonal to ω) and ∧ J,− (M ) is the subbundle of J-anti-invariant 2forms.
The fact that ω is closed implies the following identities (see [19]) where D g is the Levi-Civita connection with respect to the Riemannian metric g and X, Y, Z are any vector fields. Since N is a J-anti-invariant 2-form with values in T (M ), it follows from (2.4) that (2.5) D g JX J = −JD g X J. Moreover, we readily deduce from the relation (2.4) that the Nijenhuis tensor is identically zero if and only if ω (or equivalently J) is D g -parallel.
From the exterior derivative d, we can define the twisted exterior differential d c = (−1) p JdJ acting on p-forms (in particular d c f = Jdf for a smooth function f ). A direct computation shows the following relation for any smooth function f It follows that the almost-complex structure J is integrable if and only if d and d c anticommute.
We denote by Λ ω the contraction by the symplectic form ω, defined for a p-form ψ by Λ ω (ψ) = 1 2 2n i=1 ψ (e i , Je i , · · · ) , where {e i } is a local J-adapted orhonormal frame. As noticed by Gauduchon [16] and Merkulov [23], the commutator of Λ ω and d is equal to where δ c = (−1) p Jδ g J is the twisted codifferential acting on p-forms; here δ g is the codifferential defined as the formal adjoint of d with respect to the metric g (δ g also stands for the adjoint of D g with respect to g when it is applied to sections of ⊗ p T * (M )). The operator Λ ω commutes with J, so the relation (2.7) implies It follows from (2.8) that on any almost-Kähler manifold we have [16] (2.9) δ g d c + d c δ g = 0.
2.2. The hermitian connection. The canonical hermitian connection ∇ corresponding to J is defined by The connection ∇ preserves J (i.e. ∇J = 0) and has a J-anti-invariant torsion. It also induces the Cauchy-Riemann operator∂ on T 1,0 (M ) [16], where we recall (∂Y 1,0 )(X 0,1 ) := X 0,1 , Y 1,0 1,0 . Indeed, Proposition 2.1. For any vector fields X, Y Proof. Using the above defintion of ∇ and the relation (2.5), we have On the other hand, The proposition follows. . We consider also ρ * = R(ω), the image of the symplectic form ω by the (Riemannian) curvature operator R. The 2-form ρ * is called the * -Ricci form. We have the following relation between these Ricci-type tensors (see [5]) In the Kähler case (i.e. when D g ω = D g J = 0), we readily deduce from the relation (2.10) that ρ ∇ = ρ * , which is also equal to the usual Ricci form. Note that neither ρ ∇ nor ρ * is J-invariant in general. In fact, by (2.10) and (2.5), ρ ∇ is J-invariant if and only if ρ * is.

Holomorphic vectors fields on an almost-Kähler manifold.
In this section, we study (pseudo-)holomorphic vector fields on an almost-Kähler manifold On an almost-Kähler manifold (M 2n , ω, J, g), X is a holomorphic vector field if and only if Proof. For any vector field X, we have On the other hand J [D g X, J] = 2(D g X) − . The equality (2.11) followns from (2.5).
where {e i } is a local J-adapted orhonormal frame, δ g the formal adjoint of the Levi-Civita connection D g with respect to the metric g and ♯ stands for the isomorphism between T * (M ) and T (M ) induced by g −1 .
Proof. By using the fact that Corollary 2.4. Let α be a 1-form such that X = α ♯ is holomorphic vector field. Then Proof. Combining Lemmas 2.2 and 2.3 and using the relations (2.3) and (2.10), we have Corollary 2.5. Let (M 2n , ω, J, g) be a compact almost-Kähler manifold. Suppose that the tensor (ρ * ) + (·, J·) is negative-definite. Then, there is no non-trivial holomorphic vector field on M .
Proof. Let α be the dual (by the metric g) of a holomorphic vector field X. By Lemma 2.3, we have Using Lemma 2.2, we simplify the second term of the right hand side of (2.12) as in the proof of Corollary 2.4 Integrating (2.12), we obtain

The corollary follows
Lemma 2.6. Let (M 2n , ω, J, g) be an almost-Kähler manifold. If X is a hamiltonian Killing vector field, then where f X is the momentum of X with respect to ω (i.e. ω(X, ·) = −df X ) and ∆ g is the Riemannian Laplacian with respect to g.
Proof. By hypothesis, Combining this with Corollary 2.4 applied for the holomorphic vector field X, we obtain On the other hand (by Lemma 2.
Proof. Suppose that X = (d c f ) ♯ is a hamiltonian Killing vector field. By Lemma 2.6, we have By integrating, we obtain 1 If ∆ g f is identically zero on a compact Riemannian manifold, f must be constant and thus X = 0. The corollary follows.

2.5.
A localization formula. Let (M 2n , ω) be a compact symplectic manifold endowed with a hamiltonian S 1 -action. Let X be the generator of this action with a momentum f X i.e. ω(X, ·) = −df X . Thus, X is a hamiltonian Killing vector field with respect to some compatible almost-Kähler metrics. The fixed points of the action form a finite union of connected symplectic submanifolds [22] Then, we have the following formula (see [10,22]) for every h ∈ C. Here, c 1 (·) denote the first Chern class and we take the formal inverse 1

Extremal almost-Kähler metrics
For the rest of the paper, (M 2n , ω) is a compact and connected symplectic manifold. We denote by [ω] the deRham cohomology class of ω. Any ω-compatible almost-complex structure is identified with the induced Riemannian metric.
3.1. Hermitian scalar curvature as a moment map. We define the hermitian scalar curvature s ∇ of an almost-Kähler structure (ω, J, g) as the trace of ρ ∇ with respect to ω, i.e.
or, in equivalent way, Denote by AK ω the Fréchet space of ω-compatible almost-complex structures. The space AK ω comes naturally equipped with a formal Kähler structure (Ω, J) (described first by Fujiki in [12]). More precisely, the tangent space T J (AK ω ) at a point J is identified with the space of g-symmetric, J-anti-invariant fields of endomorphisms of T (M ) (where (ω, J, g) is the corresponding almost-Kähler metric). Then, for A, B ∈ T J (AK ω ), the Kähler form Ω is given by Let Ham(M, ω) be the group of hamiltonian symplectomorphisms of (M 2n , ω). The Lie algebra of Ham(M, ω) is identified with the space of smooth functions on M with zero mean value and the Poisson bracket; it is also equipped with an equivariant inner product, given by the L 2 -norm with respect to ω n n!
A key observation, made by Fujiki [12] in the integrable case and by Donaldson [8] in the general almost-Kähler case, asserts that the natural action of Ham(M, ω) on AK ω is hamiltonian with momentum given by the hermitian scalar curvature s ∇ . More precisely, the moment map is where s ∇ is the hermitian scalar curvature of the induced almost-Kähler metric (ω, J, g). The square-norm of the hermitian scalar curvature defines a functional on AK ω The functional (3.4) corresponds to the square-norm function of the moment map. This observation was used [5] to deduce that the critical points of (3.4) are precisely the almost-Kähler metrics g for which the symplectic gradient of their hermitian scalar curvature grad ω s ∇ , defined as the symplectic dual of ds ∇ (i.e. −ds ∇ = ω(grad ω s ∇ , ·) = g(Jgrad ω s ∇ , ·) and thus grad ω s ∇ = Jgrad s ∇ ), is a Killing vector field of g; as it is hamiltonian, this is also equivalent to being holomorphic with respect to J. Indeed, it follows from [11] that a point x 0 is critical for the square norm function of the momentum map if and only if the image of x 0 by the moment map belongs to the Lie subalgebra corresponding to the stabilizer of x 0 by the action (where the Lie algebra is identified with its dual vector space via the inner product). In our context, this precisely means that grad ω s ∇ is a Killing hamiltonian vector field. Proposition 3.2. A metric g is a critical point of (3.4) if and only if grad ω s ∇ is a Killing vector field with respect to g.

Proof.
We reproduce here a direct verification made by Gauduchon [16] using Mohsen formula [24]. The Mohsen formula states that, for a path J t ∈ AK ω , the first variation of the connection 1-forms α t of the hermtian connections on K −1 Jt (M ), induced by the canonical hermitian connections ∇ t corresponding to J t , is given by where δ gt is the codifferential with respect to the metric g t (·, ·) = ω(·, J t ·) anḋ J = d dt J t . Therefore, by definition, d dt ρ ∇t = − 1 2 dδ gtJ . Hence, by (3.2) and (2.7), we obtain d dt s ∇t = −Λ ω dδ gtJ = δ c t δ gtJ = −δ gt J t δ gtJ ; here δ c t is the twisted codifferential with respect to J t . Therefore, Using (3.5), we compute the first derivative of (3.4) in the direction ofJ where d c t is the twisted exterior differential corresponding to J t . However,J = d dt J t is a g t -symmetric, J t -anti-invariant endomorphism of T (M ). Hence, J is a critical point if and only if the symmetric, J-anti-invariant part of D g d c s ∇ is identically zero. On the other hand, for any vector field X preserving ω, we have It follows that the symmetric part of D g d c s ∇ is already J-anti-invariant. Hence, J is a critical point if and only if D g d c s ∇ is skew-symmetric which means that the symplectic gradient grad ω s ∇ is a Killing vector field.  [22], in the space of ω-compatible integrable almost-complex structures K ω .

3.2.
The extremal vector field. We fix a compact group G in the (infinite dimensional) group Ham(M, ω) of hamiltonian symplectomorphisms of (M 2n , ω). Denote by g ω ⊂ C ∞ (M ) the finite dimensional space of smooth functions which are hamiltonians with zero mean value of elements of g = Lie(G). It is well-known that g ω has a Lie algebra structure given by the Poisson bracket and g ω is isomorphic to g. Denote by Π ω the L 2 -orthogonal projection of a smooth function on g ω with respect to the volume form ω n n! . Let AK G ω be the space of ω-compatible G-invariant almost-complex structures. As AK ω is contractible, it is also connected; the same is true for AK G ω by taking the average of a path of metrics in AK ω over G. In this context, the following remark, generalizing Lemma 2 in [4] suggests the definition of an extremal vector field of AK G ω : Lemma 3.4. Let J t be a smooth family of almost-complex structures compatible with the fixed symplectic form ω, which are invariant under a compact group G of symplectomorphisms acting in a hamiltonian way on the compact symplectic manifold (M 2n , ω). Then, the L 2 -orthogonal projection of the hermitian scalar curvature s ∇t of (ω, J t , g t ) to g ω is independent of t.
Proof. By definition, any f ∈ g ω defines a vector field X = grad ω f which is in g, and is therefore Killing with respect to any of the metrics g t (·, ·) = ω(·, J t ·) in AK G ω . To prove our claim, we have to show that M f s ∇t ω n n! is independent of t. Using the relation (3.5), we obtain The fact that X = grad ω f is Killing implies that D gt d c t f is an anti-symmetric tensor. The result follows if we recall thatJ is a g t -symmetric endomorphism of T (M ).
Remark 3.5. The above lemma can also be viewed as a consequence from the fact that s ∇ is the momentum map for the action of Ham(M, ω). Indeed, consider a Lie subgroup G ⊂ H where H is a Lie group (equipped with a bi-invariant metric) acting in a hamiltonian way on a symplectic manifold with moment map µ H . Let N be a G-invariant connected subspace and denote by µ G the projection (with respect to the inner product) of the image of µ H on the dual of the Lie subalgebra of G. Since N is G-invariant, the differential of µ G | N (restriction of µ G to N ) is zero. Therefore, µ G | N is constant. In our case, H = Ham (M, ω), G = G, Lie(G) ∼ = g ω , N = AK G ω and µ H is given by (3.3). Lemma 3.4 is equivalent to the fact that µ G | N is constant.
Given any J ∈ AK G ω , we define z G ω := Π ω s ∇ , where s ∇ is the hermitian scalar curvature of (ω, J, g). This z G ω is independant of J by Lemma 3.4. Let G = T be a maximal torus in Ham(M, ω). We obtain the following lemma Lemma 3.6. For any J ∈ AK T ω , the almost-Kähler metric (ω, J, g) is extremal if and only ifs ∇ = z T ω , wheres ∇ is the integral zero part of the hermitian scalar curvature s ∇ of (ω, J, g) given bys ∇ = s ∇ − M s ∇ ω n M ω n . Proof. Let g be such an extremal metric, then X = grad ω s ∇ is a Killing field which is invariant by T . Denote by Ξ = span{X, t} where t = Lie(T ), then the closure of {exp Ξ} in the (compact) isometry group is a compact torus which contains T . By the maximality of the torus, we have X ∈ t. The other direction is obvious.
Definition 3.7. The vector field Z T ω := grad ω z T ω is called the extremal vector field relative to T .
On the other hand, as the vector field X t is T -invariant, (Φ t ) * (Z T ωt ) = Z T ωt . It follows that the introduced extremal vector field Z T ω coincides with the one defined by Futaki and Mabuchi in the Kähler case [14]. Indeed, in a fixed Tinvariant Kähler class, any two Kähler forms ω 1 and ω 2 are isotopic. Therefore, Denote by M T ω the set all T -invariant almost-Kähler metrics induced by Tinvariant symplectic forms isotopic to ω. The extremal vector field Z T ω is an obstruction to the existence of metrics of constant hermitian scalar curvature in M T ω . We recall that the space M T ω is related (via Moser's lemma) to the space AK T ω .
Corollary 3.9. If M T ω contains a metric with constant hermitian scalar curvature, then Z T ω = 0. Conversely, If Z T ω = 0, any extremal metric in M T ω is of constant hermitian scalar curvature.
Proof. We readily deduce from the defintion of Z T ω the first assertion of the corollary. Now, if Z T ω = 0, then z T ω = 0 for any ω isotopic to ω. In particular, any extremal metric in M T ω is of constant hermitian scalar curvature. 3.3. Periodicity of the extremal vector field. In this section, we show the periodicity of the introduced vector field Z T ω relative to a fixed maximal torus T in Ham(M, ω) when ω 2π is integral modulo torsion. Theorem 3.10. Assume that ω 2π is integral modulo torsion. Then, there exist a positive integer ν such that exp(2πνZ T ω ) = 1. Proof. As in the previous section, we denote by t ω ⊂ C ∞ (M ) the finite dimensional space of smooth functions which are hamiltonians with zero mean value of elements of t = Lie(T ).
In the symplectic context, Futaki and Mabuchi defined in [15] a bilinear symmetric form Φ : where f X , f Y ∈ t ω momentums of X, Y . By hypothesis, ω 2π is an integral cohomology class modulo torsion. Under this condition, Futaki and Mabuchi showed [15] that if X, Y are generators of S 1 -actions (i.e. exp (X) = exp (Y ) = 1), then Φ(X, Y ) ∈ Q. To prove our claim, we have to show that 2πΦ(X i , Z T ω ) ∈ Q for all 1 i k, where Z T ω is the extremal vector field relative to T and X 1 , · · · , X k are generators of the torus action. This would imply 2πZ T ω ∈ k i=1 QX i , so exp 2πνZ T ω = 1 for some positive integer ν. To show our claim, we recall Nakagawa's modified version [25] of Tian's formula [27] (n + 1)!2 n−1 n! 2π where f is a smooth function on M ,s ∇ is the zero integral part of the hermitian scalar curvature s ∇ of (ω, J, g), ρ ∇ is the hermitian Ricci form, µ = M ρ ∇ ω n−1 M ω n , N is an integer and ∆ g is the Riemannian Laplacian with respect to g. The formula (3.7) takes in account the normalization (3.1) and is a direct consequence of the fact that M (∆ g f ) ω n = 0 and the identities n j=0 (−1) j n j (n − 2j) k = 0 if k < n or k = n + 1, n j=0 (−1) j n j (n − 2j) n = 2 n n!.
As Φ is non-degenerate and 2πΦ(X, Z T ω ) = 1 2π M f X z T ω ω n (2π) n = 1 2π M f Xs∇ ω n (2π) n , we thus reduced the problem to show that the right hand side of (3.7) is rational when f = f X ∈ t ω is the momentum with respect to ω of a hamiltonian vector field X generating an S 1 -action. We have then essentially two integrals in the right hand side of (3.7) which are M f X ω n and 1 2π M . In order to compute the latter integral, we will use the localization formula (2.16). Since M is compact, we consider an integer N large enough such that ω is symplectic. By Lemma 2.6, the vector field X verifies On the other hand, Futaki and Mabuchi showed [15] that if M f X ω n = 0 then f X (p) ∈ 2πQ for any fixed point p ∈ M of the S 1 -action. Moreover, we have ∆ g f X (p) = −tr D g df X p = tr (JD g X) p . Note that (JD g X) p is a symmetric hermitian endomorphism which is the generator of the induced linear S 1 -action on T p (M ) (see e.g. [22]). This implies that the trace tr (JD g X) p ∈ 2πZ and therefore ∆ g f X (p) ∈ 2πZ. By (2.16) and using the power series of the exponential function in h, we obtain This concludes the proof.

The Hermitian-Einstein condition.
Definition 3.11. An almost-Kähler metric (ω, J, g) is called Hermitian-Einstein if the hermitian Ricci form ρ ∇ is a (constant) multiple of the symplectic form ω, i.e.
so the hermitian scalar curvature s ∇ is constant. Proof. Let ( ω, J, g) be an extremal almost-Kähler metric in M T ω . By Corollary 3.9 and since Z T ω = 0, we deduce that the hermitian scalar curvature s ∇ of ( ω, J, g) is constant. To prove our claim, it is enough to show that ρ ∇ is co-closed (and therefore harmonic), i.e. δ g ρ ∇ = 0, where δ g is the codifferential with respect to g. Indeed, as ρ ∇ and ω be then two harmonic forms representing the same cohomology class up to a multiple, Hodge theory implies there are equal up to the same multiple.

Explicit formula of the hermitian scalar curvature
Let (M 2n , ω, J, g) be an almost-Kähler manifold. By Darboux theorem, there exist coordinates {z i , t i } defined on an open set U such that the symplectic form ω has the form ω = n i=1 dz i ∧ dt i on U ; {z i , t i } are called the Darboux coordinates. In this section, we give an explicit formula of the hermitian scalar curvature s ∇ of (ω, J, g) in terms of the coordinates {z i , t i }. On U , the metric g is of the form, where G = (G ij ), H = (H ij ) are symmetric positive-definite matrix-valued functions which satisfy the compatibility conditions GH − P 2 = Id and HP = t P H (where t P denote the transpose of P = (P ij )). We define a local section Φ of the anti-canonical bundle ∂ti and ♭ stands for the isomorphism between T (M ) and T * (M ) via g. Let φ and ψ the real n-forms such that Φ = φ + √ −1ψ (in fact ψ and φ are related in the following way ψ(JX 1 , · · · , X n ) = φ(X 1 , · · · , X n )). We still denote by ∇ the hermitian connection induced on K −1 J (M ) by the canonical hermtian connection. For any vector field X, we have for some real 1-forms a, b (we can deduce that a = 1 2 d ln |φ| 2 ). Moreover, the hermitian Ricci form ρ ∇ is given by ρ ∇ = db. Now, span {K 1 , · · · , K n } is Lagrangian. Hence, Φ(K 1 , · · · , K n ) = φ(K 1 , · · · , K n ) = det g(K i , K j ) = det H which implies that In particular, Using Proposition 2.1, we can compute, via (4.1), the imaginary part of B and therefore ρ ∇ and s ∇ . If we denote by H −1 = (H ij ) the inverse of H = (H ij ) and G ij,k = ∂Gij ∂z k , G ,k ij = ∂Gij ∂t k etc, we obtain (details of calculations are left) 4.1. The toric case. A toric symplectic manifold (M 2n , ω) is a symplectic manifold equipped with an effective hamiltonian action of an n-dimensional torus T . It is generated by a family of hamiltonian vector fileds {K 1 , · · · , K n } which are linearly independant on a dense open set M • and satisfy the condition ω(K i , K j ) = 0 for all i, j. It is well-known that the symplectic form ω has the following local expression ω = n i=1 dz i ∧ dt i , where z i is the momentum coordinate and t i is a local coordinate such that K i = ∂ ∂ti . So, any ω-compatible T -invariant almost-Kähler metric g has the local expression An almost-Kähler manifold (M 2n , ω, g) is called locally toric if in local coordinates the symplectic form is written as ω = n i=1 dz i ∧ dt i and the almost-Kähler metric g has the form (4.4).
We deduce from (4.2) and (4.3) the expressions of the hermitian Ricci form ρ ∇ and the hermitian scalar curvature s ∇ generalizing the formula of s ∇ given by Abreu [1] in the integrable case and rediscovering the expression found by Donaldson [9]. Now, we suppose that the (lagrangian) g-orthogonal distribution to the T -orbits is involutive. This condition is automatically satisfied in the Kähler case. Then, using Frobenius' theorem, there exist local coordinates {t i } such that {dt 1 , · · · , dt n } spann the annihilator of the orthogonal distribution to the T -orbits and ω = n i=1 dz i ∧ dt i . Any such ω-compatible T -invariant almost-Kähler metric (ω, g) has the local expression Therefore, the expression (4.5) of ρ ∇ further simplifies  , defined for a sufficiently small ǫ, such that ρ ∇ǫ = ρ ∇0 for all ǫ; here ∇ ǫ is the canonical hermitian connection corresponding to J ǫ . Moreover, if (ω, J 0 , g 0 ) is Kähler, we obtain an infinite dimensional family of non-integrable ω-compatible almost-Kähler metrics . (f ik (z 1 )h ik (z 2 )) ,kj = 0, which reduces to the relations f ′ 12 = αf 22 , f ′ 11 = βf 12 , h ′ 12 = −βh 11 and h ′ 22 = −αh 12 for some (real) constants α, β which we assume non-zero. Choosing f 11 and h 22 arbitrary, the above relations determine the remaining functions f 12 , f 22 , h 11 , h 12 .
In particular, for f 11 and h 22 with compact support on V , U ij has a compact support on V . This ensures that for a sufficiently small ǫ, H ǫ = (H ǫ ij ) is positive-definite. Letting G ǫ = (H ǫ ) −1 , we obtain the ω-compatible metric elsewhere. One can check directly that for a generic choice of f 11 and h 22 , the corresponding almost-complex structure J ǫ is non-integrable. Examples 4.4. The corollary applies to the toric Kähler-Einstein surfaces CP 2 , CP 1 × CP 1 and CP 1 #3CP 2 , but also to the locally symmetric Hermitian-Einstein spaces CH 2 /Γ and CH 1 × CH 1 /Γ.
where W ± are symmetric trace-free endomorphism of ∧ ± (M ) respectively acting trivially on ∧ ∓ (M ) and r 0 is the (symmetric) operator defined on ∧ 2 (M ) as r 0 (X ∧ Y ) = r 0 (X, ·) ∧ Y + X ∧ r 0 (Y, ·); here r 0 = r − s 4 g is the trace-free part of the Ricci tensor r which is the trace of the Riemannian curvature R and s is the Riemannian scalar curvature defined as the trace of the Ricci tensor (for more details see [3]). The tensor W + (resp. W − ) is called the selfdual Weyl tensor (resp. anti-selfdual Weyl tensor).
We   Moreover, the equality holds in (4.9) if and only if g is an extremal almost-Kähler metric with negative constant hermitian scalar curvature s ∇ and at each point ω is an eigenform of W + correponding to its lowest eigenvalue. For the Kähler-Einstein metric on CH 2 /Γ and CH 1 × CH 1 /Γ, s = s ∇ is negative and thus ω belongs to the lowest eigenspace of W + . Then, by Corollary 4.3, we obtain an infinite dimensional family of non-integrable Hermitian-Einstein almost-Kähler metrics whose the almost-Kähler form belongs to the lowest eigenspace of W + with non-positive constant hermitian scalar curvature s ∇ . Other examples of non-integrable almost-Kähler metrics saturating the inequality (4.9) appear in [26].

further remarks and questions
(1) In [9], it is shown that in the toric case the existence of an extremal almost-Kähler metric is closely related (and conjecturally equivalent) to the existence of an extremal Kähler metric; this was further generalized to certain toric bundles in [4]. It will be interesting to establish a similar link in general, assuming that there are integrable complex structures in AK T ω . (2) One expects that there will be infinite families of extremal almost-Kähler metrics (should they exist) in AK T ω , thus generalizing the results of [18] and Example 4.4. Indeed, this is suggested from the GIT formal picture in [8], where the existence and uniqueness (modulo the action of Ham(M, ω)) of the extremal almost-Kähler metrics are expected to hold within a 'stable' complexified orbit for the action of Ham(M, ω) on AK ω . However, as Ham(M, ω) does not have a nature complexification, the description of the complexified orbits is not obvious and is clearly established only on Kähler manifolds with H 1 (M, R) = {0} for the action of Ham(M, ω) on the subset K ω of integrable, ω-compatible almost-complex structures: In this case, the complexified orbit of J ∈ K ω is identified in [8] (via Moser's lemma) with the space of Kähler metrics within the Kähler class Ω = [ω] on (M, J). In the absence of holomorphic vector fields on (M, J), the stability theorem of [21] and [13] implies that the existence of an extremal Kähler metric in a given complexified orbit is an open condition on the space of such orbits; this was generalized in [4], by fixing a maximal tourus T ⊂ Aut 0 (M, J) ∩ Ham(M, ω) (where Aut 0 (M, J) is the connected component of the finite dimensional group of complex automorphisms of (M, J)) and considering T -invariant ω-compatible complex structures. We expect the openess result to generally hold in the non-integrable case, by considering suitable complexified orbits in AK T ω . (3) A well-know result of Calabi states that any extremal Kähler metric (ω, g) on a compact complex manifold (M, J) is invariant under a maximal connected compact subgroup G of Aut 0 (M, J) ∩ Ham(M, ω). It will be desirable to know whether or not such a G (or a maximal torus in it) is also maximal as a compact subgroup of Ham(M, ω)? More generally, one would like to know whether or not an extremal almost-Kähler metric is necessarily invariant under a maximal torus in Ham(M, ω)?
(4) It would be interesting to know how does the extremal vector field Z T ω , which we introduced as an invariant of a maximal torus T in Ham(M, ω), characterize this torus up to congugacy in Symp(M, ω) or Ham(M, ω); elaborating on the theory appearing in recent works [2,17] would be an appealing direction of further investigation.