The General Definition of the Complex Monge-Amp\`ere Operator on Compact K\"ahler Manifolds

TL;DR

This paper defines a broad class of quasi-plurisubharmonic functions on compact Kähler manifolds where the complex Monge-Ampère operator is well-defined and convergent, extending previous classes like Cegrell's.

Contribution

It introduces the class ${ m extbf{F}}(X,oldsymbol{ extomega})$, proving its convexity and inclusion of Cegrell's class, thus broadening the scope of the Monge-Ampère operator.

Findings

Defined a new subclass ${ m extbf{F}}(X,oldsymbol{ extomega})$ for the Monge-Ampère operator.

Proved ${ m extbf{F}}(X,oldsymbol{ extomega})$ is convex.

Showed ${ m extbf{F}}(X,oldsymbol{ extomega})$ contains all functions in the Cegrell class.

Abstract

We introduce a wide subclass ${\cal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact K\"ahler manifold, on which the complex Monge-Amp\`ere operator is well-defined and the convergence theorem is valid. We also prove that ${\cal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class.