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

Bin Guo, Duong H. Phong, Freid Tong, Chuwen Wang

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.

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 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.
Paper Structure (3 sections, 4 theorems, 38 equations)

This paper contains 3 sections, 4 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The Monge-Ampère equation
  3. Complex Hessian equations

Key Result

Theorem 1

Consider the equation (eqn:MA), and assume that the cohomology class of $\chi$ is nef. For any $s>0$, let $\Omega_s = \{\varphi_t - V_t \leq -s\}$ be the sub-level set of $\varphi_t - V_t$. (a) Then there are constants $C=C(n, \omega, \chi)>0$ and $\alpha_0 = \alpha_0(n,\omega,\chi)>0$ such that where $A_s = \int_{\Omega_s} (-\varphi_t + V_t - s) e^F \omega^n$ and $E_t = \int_X (-\varphi_t + V_t)

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Lemma 2