Let $X$ be a complex algebraic variety over $\mathbf C$, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.

Question. Is $\pi_1(X)$ actually a subgroup of the group of isometries of $\tilde X$ with respect to the Kahler-Einstein metric?

I suspect it is, but I don't know enough about analytic geometry to be completely sure.

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.

Question. Assume $X$ is compact. Is $\pi_1(X)$ actually a subgroup of the group of isometries of $\tilde X$ with respect to the Kahler-Einstein metric?

I suspect it is, but I don't know enough about analytic geometry to be completely sure.