Compact almost Kähler manifolds with divergence-free Weyl conformal tensor (Q1770911)

Let \((M^{2n},J,g)\) be an almost Hermitian manifold, where \(J\) is an almost complex structure, and \(g\) is a \(J\)-invariant metric on \(M\). \((M,J,g)\) is an almost Kähler manifold if the Kähler form is closed. Kähler manifolds are almost Kähler manifolds. However, in general the converse is not true. Let \(W\) be the Weyl conformal tensor, \(\delta W(Z,X,Y)=-\text{Trace}_g\nabla W(\cdot,Z,X,Y)\), where \(\nabla\) is the Levi-Civita connection of \(g\). \(J\) acts on the bundle of 2-forms \(\Lambda^2\) as an involutive endomorphism. Let \(\Lambda^{J+}\), \(\Lambda^{J-}\) be eigen-subbundles of eigenvalues \(\pm1\), and let \(\pi_2:\,\Lambda^1\otimes\Lambda^2\rightarrow\Lambda^1\otimes\Lambda^{J-}\) be the projection. The author proves that \((M,J,g)\) is a Kähler manifold assuming that \((M,J,g)\) is a compact \(2n\)-dimensional almost Kähler manifold whose Ricci tensor is positive semi-definite, and whose Weyl conformal tensor \(W\) satisfies the equation \(\pi_2(\delta W)=0\).