The Monge-Amp\`ere equation for (n-1)-plurisubharmonic functions on a compact K\"ahler manifold

TL;DR

This paper proves the solvability of a Monge-Ampère equation for (n-1)-plurisubharmonic functions on compact Kähler manifolds, leading to new Calabi-Yau theorems for various special metrics.

Contribution

It introduces a method to solve the Monge-Ampère equation for (n-1)-plurisubharmonic functions on compact Kähler manifolds, extending Calabi-Yau results.

Findings

Existence of solutions to the Monge-Ampère equation for (n-1)-plurisubharmonic functions.

Application to Calabi-Yau theorems for balanced, Gauduchon, and strongly Gauduchon metrics.

Extension of classical results to a broader class of complex geometric structures.

Abstract

A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.