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

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