Complex structures on quaternionic manifolds
In the first part of this work we consider compact Riemannian manifolds \(M\) with holonomy in \(Sp(n)Sp(1)\). We show that \(M\) admits a compatible complex structure if and only if the holonomy is in \(Sp(n)\), up to finite coverings. We also show that the sign of the Ricci curvature completely determines the algebraic dimension of the twistor space. In the second part, by way of contrast, we give two geometric constructions of simply-connected quaternionic manifolds with a compatible complex structure which is not hypercomplex. The first examples are non-compact and symmetric but we can show existence of compact quotients in some special dimension. The second one is compact and follows from general results of \textit{D. D. Joyce} [J. Differ. Geom. 35, No. 3, 743-762 (1992; Zbl 0782.53062)].
scientific article; zbMATH DE number 609684