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Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the intersection cohomology of any projective complex variety, in particular of $X$, carries a natural pure Hodge structure.

Is it true that it satisfies $$H^{p,q}=0 \mbox{ if } p\ne q?$$

This is known to be the case if $X$ is smooth.

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    $\begingroup$ I am tempted to argue as follows: $X$ has a toric resolution $Y\rightarrow X.$ By the decomposition theorem, the intersection cohomology of $X$ is a direct summand of the cohomology of $Y$, and as you note, the cohomology of $Y$ is Hodge-Tate. $\endgroup$
    – dhy
    Commented Jan 10, 2021 at 8:55

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