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Let $X$ be a normal algebraic variety over $\mathbb{C}$. What is the relationship between:

-$X$ has rational singularities, i.e. for any resolution of singularities $f:\tilde{X} \to X$, we have $R^*f_*\mathcal{O}_{\tilde{X}} \cong \mathcal{O}_X$.

-$X$ is rationally smooth, i.e. $IC(X)$ is isomorphic to the constant sheaf up to a certain shift (you can assume that $X$ is equidimensional).

I'm particularly interested in the case where $X$ is locally complete intersection.

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  • $\begingroup$ All toric varieties have rational singularities, but most are not rationally smooth. $\endgroup$ Commented Jan 30 at 6:44

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