The Monge-Ampère equation for $(n-1)$-quaternionic PSH functions on a hyperKähler manifold

Abstract

We prove the existence of unique smooth solutions to the quaternionic Monge-Ampère equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperKähler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperKähler metric comparing to previous results [14].

31 pages