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Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K_X$.

Does there always exist a Kähler-Einstein metric on $X$?

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Yes, this is a result of Essydieux-Guedj-Zeriahi: "Singular Kähler-Einstein metrics" J. Amer. Math. Soc. 22 (2009), 607-639.

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