$L^1$-Stability for complex Monge-Amp\`ere equations

We first establish the weak stability results for solutions of complex Monge-Amp\`ere 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-Amp\`ere 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.