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

TL;DR

This paper proves the solvability of the quaternionic Monge-Ampère equation on hyperKähler manifolds, confirming a conjecture by Alesker and Verbitsky, and extends Calabi-Yau type results to HKT metrics without flatness assumptions.

Contribution

It provides the first proof of the conjecture for hyperKähler with torsion manifolds without assuming flatness of the hypercomplex structure.

Findings

Solves the quaternionic Monge-Ampère equation on hyperKähler manifolds.

Establishes a Calabi-Yau type theorem for HKT metrics.

Confirms the Alesker-Verbitsky conjecture under broad conditions.

Abstract

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.