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The Kahler-Einstein metric is an example of a canonical metric on a Kahler manifold. We say that a metric $\omega$ is Kahler-Einstein if $Ric(\omega)=\lambda\omega$, where $\lambda\in\{-1,0,+1\}$.

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CSC Kahler metrics on a blown-up torus

This statement about blow ups of torus is not correct. Take any aglebraic $2$-torus with a smooth curve $C$ of genus $>1$. Blow up $C^2+1$ points on $C$ and apply Theorem 1 here: https://arxiv.org/pdf …
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