$L^1$-Stability for complex Monge-Ampère equations

We first establish the weak stability results for solutions of complex Monge-Ampère equations in relative full mass classes, extending the results known to hold in the full mass class. Building on weak stability, we then prove the ${\mathcal{C}}^{k,\alpha }$ stability of solutions to complex Monge-Ampère equations on quasi-projective varieties. As an application, we study the limit of the singular Ricci-flat metrics on $\mathbb{Q}$-Calabi-Yau projective varieties, inspired by Tosatti’s work on Calabi-Yau projective manifolds.