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We study families of complex Monge-Ampère equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori L ∞ -estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the study of the Kähler-Ricci flow.doi:10.1093/imrn/rnn070 fatcat:3ucatv333fc5hmf5gny3a6uho4