Almost Kähler 4-manifolds with J-invariant Ricci Tensor and Special Weyl Tensor

for any tangent vectors X,Y to M . If the almost complex structure J is integrable we obtain a Kähler structure. Many efforts have been done in the direction of finding curvature conditions on the metric which insure the integrability of the almost complex structure. A famous conjecture of Goldberg [26] states that a compact almost Kähler, Einstein manifold is in fact Kähler. Important progress was made by K. Sekigawa who proved that the conjecture is true if the scalar curvature is non-negative [46]. The case of negative scalar curvature is still wide open, although recently there has been some significant progress in dimension 4, see [7, 8, 9, 42, 43, 44]. It is now known that if the conjecture turns out to be true, the compactness should play an essential role. Nurowski and Przanowski [43] constructed a 4-dimensional, local example of Einstein, strictly almost Kähler manifold; this was generalized by K.Tod (see [8, 9]) to give a family of such examples. It is interesting to remark that the structure of the Weyl tensor of all these examples is unexpectedly special. An important recent result of J. Armstrong [8, 9] states that any 4-dimensional almost Kähler, non-Kähler, Einstein manifold is obtained by Tod’s construction, provided that the Kähler form is an eigenform of the Weyl tensor. In the compact case, on the other hand, some of the positive partial results on the conjecture in dimension 4 have been obtained exactly by imposing some additional assumptions on the structure of the Weyl tensor ([7, 8, 9, 44]). The recently discovered Seiberg-Witten invariants could have an impact towards a complete answer to the Goldberg conjecture in dimension 4. These are invariants of smooth, oriented, compact 4-dimensional manifolds. The works of Taubes [47, 48, 49] and others showed the invariants to be particularly interesting for symplectic 4-manifolds. (See also [34] for a quick introduction to Seiberg-Witten invariants and some of their applications to symplectic geometry.) LeBrun [37], [38], [39] and Kotschick [35] have also