Dirichlet problem of complex Monge-Amp\`ere equation near an isolated KLT singularity

TL;DR

This paper addresses the Dirichlet problem for the complex Monge-Ampère equation near isolated KLT singularities, extending previous results to include boundary conditions and constructing solutions on strongly pseudoconvex domains in complex space.

Contribution

It generalizes existing solutions of the Monge-Ampère equation to settings with isolated KLT singularities and boundary conditions, providing new existence results.

Findings

Solved the Dirichlet problem near isolated KLT singularities.

Constructed solutions on strongly pseudoconvex domains in ^n.

Extended previous work to include boundary conditions.

Abstract

We solve the Dirichlet Problem of Monge-Amp\`ere equation near an isolate Klt singularity, which generalizes the result of Eyssidieux-Guedj-Zeriahi \cite{EGZ}, where the Monge-Amp\`ere equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge-Amp\`ere equation with isolated singularity on strongly pseudoconvex domain $\Omega$ contained in $\mathbb{C}^n$.