An application of a $C^2$-estimate for a complex Monge-Amp\`ere equation

TL;DR

This paper provides an alternative proof linking complex Monge-Ampère equations to the projectivity of certain compact Kähler manifolds with negative Ricci curvature and the ampleness of their canonical bundle.

Contribution

It introduces a new approach using a $C^2$-estimate for the complex Monge-Ampère equation to prove key geometric properties of Kähler manifolds.

Findings

Proof of projectivity for Kähler manifolds with $ ext{Ric}_k<0$

Establishment of ampleness of the canonical line bundle

Application of $C^2$-estimate technique in complex geometry

Abstract

By studying a complex Monge-Amp\`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact K\"ahler manifold $N^n$ with $\Ric_k< 0$ for some integer $k$ with $1<k<n$, and the ampleness of the canonical line bundle $K_N$.