Pluripotential theory on quaternionic manifolds

TL;DR

This paper extends pluripotential theory to quaternionic manifolds by defining plurisubharmonic functions and a Monge-Ampère operator, generalizing known results from hypercomplex cases with new technical tools.

Contribution

It introduces a new class of plurisubharmonic functions and a Monge-Ampère operator on quaternionic manifolds, utilizing Baston differential operators with a novel multiplicativity property.

Findings

Defined plurisubharmonic functions on quaternionic manifolds

Constructed a Monge-Ampère operator satisfying classical theorems

Developed new properties of Baston differential operators

Abstract

On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Amp\`ere operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property.