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 Eyssidieux et al. (J Am Math Soc 22(3):607–639, 2009) and Song and Tian (Inv Math 207:519–595, 2017), Tsuji (Math Ann 281(1):123–133, 1988), Tian and Zhang (Chin Ann Math Ser 27B(2):179–192, 2006) are continuous. We also provide an affirmative answer to a conjecture in Eyssidieux et al. (J Am Math Soc 22(3):607–639, 2009) by showing that a resolution of any compact normal Kähler space satisfies the smooth approximation property. Finally, we settle the continuity of the potentials of the weak Kähler–Ricci flows Guedj et al. (Geom Top 24:1225–1296, 2020) and Song and Tian (Inv Math 207:519–595, 2017) on compact Kähler varieties with log terminal singularities.