Hermitian conformal classes and almost Kähler structures on 4-manifolds (Q1807647)

The authors prove that on most compact complex surfaces which admit symplectic forms, each Hermitian conformal class contains almost Kähler metrics. The number of symplectic forms compatible with a given metric can also be described. For precise formulations of the results, the reader should see the original paper, where many motivations are also presented. As a consequence, one gets for instance the following: On blow-ups of primary Kodaira surfaces, Hermitian conformal classes do not contain almost Kähler metrics. There are also other applications. Among others, alternative proofs of the results of \textit{C. LeBrun} [Commun. Anal. Geom. 5, 535-555 (1997; Zbl 0901.53028)] about the Yamabe constants of Hermitian conformal classes are given. Moreover, some particular answers to a question of \textit{D. E. Blair} [Differ. Geom. Appl., Proc. Conf. Dubrovnik 1988, 49-58 (1989; Zbl 0691.53024)] concerning isometries of almost Kähler metrics are stated.