Complex structures on Riemannian four-manifolds
The purpose of this work is to investigate those connected oriented Riemannian 4-manifolds admitting several compatible complex structures which are independent. It is shown that \((M^4,g)\) is necessarily anti-self-dual if there exists three compatible complex structures \(J_1\), \(J_2\), \(J_3\) such that for each \(i\neq j\) there is a point \(p\in M\), where \(J_i(p)\neq\pm J_j(p)\). Moreover, if \(M\) is a compact anti-selfdual manifold admitting two compatible complex structures \(J_1\neq\pm J_2\), then, up to a conformal rescaling of the metric, it is a flat torus, a Ricci-flat Kähler K3 surface, a conformally flat Hopf surface or it is diffeomorphic to \((S^1\times S^3)\# n\overline{\mathbb{C} P^2}\), \(n\geq 1\). Also, applications to the existence of orientation-preserving isometries, Killing vector fields and hyper-Hermitian structures are given.
compatible independent complex structures
scientific article; zbMATH DE number 1058824