Hyper-Hermitian quaternionic Kähler manifolds (Q1610263)

The author investigates quaternionic Kähler manifolds which carry a hyper-Hermitian structure which trivializes the quaternionic structure, i.e. hHqK manifolds. Alekseevsky, Marchiafava and Pontecorvo have proved that such compact manifolds are locally hyper-Kähler, in particular there are no complete hHqK-manifolds with positive scalar curvature. They stated a conjecture that the only complete simply-connected hHqK manifold with non-zero scalar curvature is the quaternionic hyperbolic space \(\mathbb{H} H^n\). The author gives the partial answer to this conjecture proving that it holds if we additionally assume that the Lee form of the hyper-Hermitian structure is closed. The author also proves that the every locally symmetric hHqK manifold is locally homothetic to \(\mathbb{H} P^n\) or \(\mathbb{H} H^n\). He gives additionally a local classification of hHqK manifolds with closed Lee form proving that every such manifold is locally isometric to the Swan bundle or to the family of examples constructed by the author.