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On the Hermitian curvature of symplectic manifolds

We study the integrability of a (almost) complex structure calibrated by a symplectic form. We find new sufficent conditions.


Introduction
The interplay between complex and symplectic structures has been recently studied by many authors. Indeed, on any symplectic manifold (M, κ) there exists a κ-calibrated almost complex structure J, so that (M, g, J, κ) is an almost Kähler manifold. In the context of almost Kähler geometry it is natural to study the integrability of the almost complex structure. In [4] Goldberg proved that, if the curvature operator of an almost Kähler manifold (M, g, J, κ) commutes with J, then (M, g, J, κ) is a Kähler manifold. He also conjectured that an Einstein almost Kähler metric on a compact manifold is a Kähler metric. If the scalar curvature is nonnegative the conjecture has been proved by Sekigawa in [6]. For a survey, other references and results on this topic we refer to [2].
In this paper we show that, if (M, g, J, κ) is an almost Kähler manifold satisfying certain properties, then it is Kähler. Namely we give some conditions on the derivative of the Nijenhuis tensor and on the curvature respectively, in order that J is integrable. In section 2 we start by recalling some facts and fixing some notations. In section 3 we prove that if the (0, 1)-part of the covariant derivative of the Nijenhuis tensor of J vanishes, then J is integrable (theorem 3.3). This result generalizes theorem 2 of [7]. In section 4 we consider three types of curvature tensors on (M, g, J, κ). Namely, the Riemann curvature R, the curvature R of the Hermitian connection ∇ and the tensor R J defined by We show that if R J and R have the same components along certain directions, then J is integrable (theorem 4.2). Finally we prove that if the bisectional curvature of g and the Hermitian bisectional curvature coincides, then g is a Kähler metric (theorem 4.8).
A key tool in the proof of our results is the existence of generalized normal holomorphic frames (see [7]).

Some integrability conditions
In this section we give conditions on the covariant derivative of the Nijenhuis tensor in order that the complex structure is integrable.
Let (M, κ, J, κ) be a 2n-dimensional almost Kähler manifold. We denote by ∇ the Levi-Civita connection relatively to g and by R the curvature tensor of g.
We have the following (see [3] and [7]) We recall that the following fundamental relation holds: From the complex extension of (1) it can be easily proved the following (see [7]) The covariant derivative along (0, 1)-vector fields of the almost complex structure J is a tensor of complex type (0, 1), i.e.
As a first application of the generalized normal holomorphic frames we have the following Therefore, by the assumption ∇ ′′ N J = 0, we get Moreover a direct computation gives Hence (2) and (3) Then the Nijenhuis tensor of J is the antisymmetric part of B. In [7] (theorem 2) it is proved that an almost complex structure J is integrable if and only if ∇ ′′ B = 0 . Therefore theorem 3.2 generalizes theorem 2 of [7] .
Remark 3.5. With respect to a generalized normal holomorphic frame, {Z 1 , . . . , Z n }, the components of the curvature tensor R ikrs (o) and R ikrs (o) are given by In order to study the integrability of a κ-calibrated almost complex structure J it is useful to introduce the following tensor where B is defined by (4). A first property of the tensor L is given by the following i.e.
By a direct computation we get Then we have In a similar way we obtain The second part of the proof is straightforward.
As a corollary of theorem 3.3 we have the following (see also [7]): for any complex vector fields Z i , Z j of type (1, 0); then (M, g, J, κ) is a Kähler manifold.
Proof. Let {Z 1 , . . . , Z n } be a generalized normal holomorphic frame around o. A direct computation gives

Curvature and integrability
In this section we give an integrability condition in terms of curvature. We start defining the following tensor We have Proof. By the definition of R J it follows that By corollary 3.2, we have (∇ i J)Z j = 0 . Therefore we obtain The previous lemma and equation (5) give us . Moreover equation (7) and lemma (3.6) imply the following for any Z i , Z j of type (1, 0), then J is integrable. for any X, Y real vector fields.
Let ∇ be the Hermitian connection on (M, g, J, κ); ∇ is the connection defined by Then ∇ preserves the metric g, the almost complex structure J and its torsion is given by the Nijenhuis torsion, namely Let us denote by R the curvature tensor of ∇.
From [6] we have the following

Now we have
From corollary 3.2 we have (∇ j J)Z r ; then we get i.e. (9) Let assume now that {Z 1 , . . . , Z n } is a generalized normal holomorphic frame around a point o. Then equations (9) and (5) imply for any Z i , Z j (1, 0)-fields, then (M, g, J, κ) is a Kähler manifold. for any pair of real vector fields X, Y The last theorem can be stated in terms of the bisectional curvatures of ∇ and ∇. Let p ∈ M and v, w be two unit vector of T p M . Then the holomorphic bisectional curvature of the planes σ 1 =< v, Jv >, σ 2 =< w, Jw > is defined by K[p](σ 1 , σ 2 ) = R(v, Jv, w, Jw) .