The complex Monge-Amp\'{e}re equation on the complement of a divisor

TL;DR

This paper studies the complex Monge-Ampère equation on Kähler manifolds with cusp singularities, proving existence of classical solutions under weak regularity conditions on the right-hand side.

Contribution

It establishes the existence of classical solutions in weighted Sobolev spaces for the Monge-Ampère equation with weak regularity data on singular Kähler manifolds.

Findings

Existence of classical solutions in weighted Sobolev spaces

Solutions exist when right-hand side has regularity in weighted W^{1,p_0} spaces

Applicable to manifolds with cusp singularities

Abstract

We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some \emph{weighted} $W^{1,p_0}$ space for $p_0 > 2n$, the Monge-Amp\'{e}re equation has a classical $W^{3,p_0}$ solution.