Remarks on a four-dimensional compact almost Kähler Einstein manifold (Q1611577)

An almost Hermitian manifold \((M,J,g)\) is called an almost Kähler manifold if its fundamental 2-form \(\Omega\), defined by \(\Omega(X,Y)=g(X,JY)\), is closed. If the almost complex structure \(J\) is integrable, then \(M\) is a Kähler manifold. The Goldberg conjecture [\textit{S. I. Goldberg}, Proc. Am. Math. Soc. 21, 96-100 (1969; Zbl 0174.25002)] states that a compact Einstein almost Kähler manifold is necessarily Kähler. Although the conjecture is still open, progress has been made by imposing additional curvature conditions. \textit{K. Sekigawa} [J. Math. Soc. Japan 39, 677-684 (1987; Zbl 0637.53053)] showed that this conjecture is true if the scalar curvature is nonnegative. The case of negative scalar curvature remains still open. \textit{V. Apostolov} and \textit{J. Armstrong} [Trans. Am. Math. Soc. 352, 4501-4513 (2000; Zbl 0990.53017)] proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor (in such case, the manifold is said to have a Hermitian Weyl tensor) is necessarily Kähler. -- Let \(\text{Ric} ^*\) be the Ricci \(*\)-tensor on an almost Kähler manifold defined by \[ \text{Ric}^* (X,Y) = \frac 12 \text{trace} \{Z\to R_{XJY}JZ\}. \] \textit{T. Oguro} and \textit{K. Sekigawa} [Kodai Math. J. 24, 226-258 (2001; Zbl 1008.53055)] proved that if \((M,J,g)\) is a four-dimensional almost Kähler Einstein manifold of constant \(*\)-scalar curvature, then \(M\) is a Kähler manifold. Also, if \((M,J,g)\) is a four-dimensional compact almost Kähler Einstein manifold and the norm of the skew-symmetric part of the Ricci \(*\)-tensor is a constant, then \(M\) is a Kähler manifold. In this paper, the authors prove that if \((M,J,g)\) is a four-dimensional compact almost Kähler Einstein manifold, then the skew-symmetric part of the Ricci \(*\)-tensor vanishes somewhere on \(M\). Moreover, the authors present a new proof of the theorem by Apostolov and Armstrong.