Integrability Conditions For Almost Hermitian And Almost Kaehler 4-Manifolds

Abstract

If $W_+$ denotes the self dual part of the Weyl tensor of any K\"ahler 4-manifold and $S$ its scalar curvature, then the relation $|W_+|^2 = S^2/6$ is well-known. For any almost K\"ahler 4-manifold with $S \ge 0$, this condition forces the K\"ahler property. A compact almost K\"ahler 4-manifold is already K\"ahler if it satisfies the conditions $| W_+ |^2 = S^2/6$ and $\delta W_+=0$ and also if it is Einstein and $| W_+|$ is constant. Some further results of this type are proved. An almost Hermitian 4-manifold $(M,g,J)$ with $\mathrm{supp} (W_+)=M$ is already K\"ahler if it satisfies the condition $| W_+ |^2 = 3 (S_{\star} - S/3)^2 /8$ together with $|\nabla W_+ | = | \nabla |W_+||$ or with $\delta W_+ + \nabla \log | W_+ | \lrcorner W_+ =0$, respectively. The almost complex structure $J$ enters here explicitely via the star scalar curvature $S_{\star}$ only.