On $L^{\infty }$ estimates for Monge-Ampère and Hessian equations on nef classes

Abstract

The PDE approach developed earlier by the first three authors for $L^{\infty }$ estimates for fully non-linear equations on Kähler manifolds is shown to apply as well to Monge-Ampère and Hessian equations on nef classes. In particular, one obtains a new proof of the estimates of Boucksom-Eyssidieux-Guedj-Zeriahi and Fu-Guo-Song for the Monge-Ampère equation, together with their generalization to Hessian equations.