A priori $L^{\mathrm{\infty }}$-estimates for degenerate complex Monge-Amp\`ere equations

We study families of complex Monge-Amp`ere equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\mathrm{\infty }}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the study of the K"ahler-Ricci flow.