On $L^\infty$ estimates for Monge-Amp\`ere and Hessian equations on nef classes

TL;DR

This paper extends a PDE-based method to establish $L^ olinebreak\infty$ estimates for Monge-Ampère and Hessian equations on nef classes, providing new proofs and generalizations of existing results on Kähler manifolds.

Contribution

It applies a PDE approach to nef classes, offering new proofs and extending $L^ olinebreak\infty$ estimates to Hessian equations beyond previous Kähler-specific results.

Findings

New proof of Boucksom-Eyssidieux-Guedj-Zeriahi estimates

Extension of estimates to Hessian equations

Unified PDE approach for nef classes

Abstract

The PDE approach developed earlier by the first three authors for $L^\infty$ estimates for fully non-linear equations on K\"ahler manifolds is shown to apply as well to Monge-Amp\`ere 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-Amp\`ere equation, together with their generalization to Hessian equations.