Special metrics in hypercomplex geometry

We investigate the existence and geometric properties of special hyperhermitian metrics. First of all, we characterise hypercomplex structures with Obata holonomy in $\mathrm{SL}(n, \mathbb{H})$ in terms of the existence of quaternionic Gauduchon metrics together with the vanishing of a hypercomplex cohomological invariant. In view of this, the quaternionic Gauduchon and quaternionic balanced conditions are investigated at length: we describe their properties and determine criteria for their existence. Furthermore, we prove an incompatibility result concerning strong HKT and balanced hyperhermitian metrics, confirming an open conjecture by Fino and Vezzoni in the hypercomplex framework. Finally, we introduce an Einstein-type condition, determining basic properties, obstructions and providing examples. In particular, we show that Joyce's manifolds always admit such type of metrics.