Uniform estimates for complex Monge-Amp\`ere equations: big cohomology classes

Abstract

We prove uniform a priori estimates for solutions to degenerate complex Monge--Amp\`ere equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Amp\`ere equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on K\"ahler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Amp\`ere equations.