Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

Abstract

We prove the continuity of bounded solutions to complex Monge-Ampère equations on reduced, locally irreducible compact Kähler spaces. This in particular implies that any singular Kähler-Einstein potentials constructed in \cite{EGZ09} and \cite{Tsuji88, TianZhang06, ST17} are continuous. We also provide an affirmative answer to a conjecture in \cite{EGZ09} by showing that a resolution of any compact normal Kähler space satisfies the continuous approximation property. Finally, we settle the continuity of the potentials of the weak Kähler-Ricci flows \cite{ST17, GLZ20} on compact Kähler varieties with log terminal singularities.

20 pages. Comments are welcome. The final version is to appear in Math. Ann