A priori estimates for a generalised Monge-Amp\`ere PDE on some compact K\"ahler manifolds

TL;DR

This paper establishes a priori estimates and existence results for a generalized Monge-Ampère PDE on compact Kähler manifolds, with applications to Chern-Weil theory and mirror symmetry.

Contribution

It provides new $C^0$ and gradient estimates for a broad class of nonlinear PDEs on Kähler manifolds and introduces a method-of-continuity approach for proving existence.

Findings

Proven $C^0$ a priori estimate for the PDE.

Established gradient estimates in specific cases.

Applied results to problems in Chern-Weil theory and mirror symmetry.

Abstract

We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the K\"ahler form on a K\"ahler manifold. A $C^0$ \emph{a priori} estimate is proven in general and a gradient estimate is proven in certain cases. Independently, we also provide a method-of-continuity proof via a path of K\"ahler metrics to recover the existence of solutions in some of the known cases. Known results are then applied to an analytic problem arising from Chern-Weil theory and to a special Lagrangian-type equation arising from mirror symmetry.