Almost Kähler \(4\)-manifolds with \(J\)-invariant Ricci tensor (Q2705839)

Let \((M,\omega)\) be a compact 4-dimensional symplectic manifold. The author shows that if \(g\) is a compatible almost Kähler metric with \(J\)-invariant Ricci tensor \(\rho\) and nonnegative scalar curvature then \(g\) is a Kähler metric. The same conclusion (that \(g\) is Kähler) is shown to hold when the scalar curvature is constant (of arbitrary sign) yet additionally \((c_1\cup[\omega])(M)\geq 0\), where \(c_1\) is the first Chern class of \(M\). This generalizes -- in real dimension four -- a result of \textit{K. Sekigawa} [J. Math. Soc. Japan 39, 677-684 (1987; Zbl 0637.53053)] who provided a partial solution to the Goldberg conjecture (the almost complex structure of any compact Einstein almost Kähler manifold is integrable) when the scalar curvature is \(\geq 0\). In arbitrary dimension the same author had previously shown that a compact almost Kähler manifold is actually Kähler, provided that \(\lambda g(X,X)\rho(X)\leq 2\lambda g(X,X)\), for some \(\lambda\geq 0\) [Kodai Math. J. 18, 156-163 (1995; Zbl 0836.53028)]. For related results see \textit{V. Apostolov} and \textit{T. Draghici} [Q. J. Math. 51, 275-294 (2000; Zbl 0983.53047) (see the following review) and Differ. Geom. Appl. 11, 179-195 (1999; Zbl 0978.53117)]. For an example underlying the importance of the compactness assumption see \textit{P. Nurowski} and \textit{M. Przanowski} [Class. Quantum Gravity 16, L9--L13 (1999; Zbl 0979.53045) (cf. also \textit{T. Sato} [Some examples of almost Kähler 4-manifolds, Balkan J. Geom. Appl. 5, 113-137 (2000; Zbl 0986.53024)] for related examples).