On the Alesker-Verbitsky conjecture on hyperK\"ahler manifolds

We solve the quaternionic Monge-Amp`ere equation on hyperK"ahler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperK"ahler with torsion manifold, at least when the canonical bundle is trivial holomorphically. The novelty in our approach is that we do not assume any flatness of the underlying hypercomplex structure which was the case in all the approaches for the higher order a priori estimates so far. The resulting Calabi-Yau type theorem for HKT metrics is discussed.