Continuity method with movable singularities for classical Monge-Amp\`ere equations

TL;DR

This paper investigates the strong continuity of solutions to complex Monge-Ampère equations with prescribed singularities on compact Kähler manifolds, introducing new conditions and methods for stability and openness related to Kähler-Einstein metrics.

Contribution

It introduces a novel continuity method with movable singularities for Monge-Ampère equations and establishes conditions for strong continuity and stability of solutions.

Findings

Established strong continuity conditions for solutions with prescribed singularities.

Proved an openness result for decreasing singularities in Monge-Ampère equations of Fano type.

Deduced stability results for (log-)Kähler Einstein metrics under modifications.

Abstract

On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp\`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the strong continuity of solutions when the right-hand sides are modified to include all (log) K\"ahler-Einstein metrics with prescribed singularities. Our findings can be interpreted as closedness of new continuity methods in which the densities vary together with the prescribed singularities. For Monge-Amp\`ere equations of Fano type, we also prove an openness result when the singularities decrease. As an application, we deduce a strong stability result for (log-)K\"ahler Einstein metrics on semi-K\"ahler classes given as modifications of $\{\omega\}$.