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Intersection complexes of toric varieties and mixed Hodge modules

We prove the structure theorem of the intersection complexes of toric varieties in the category of mixed Hodge modules. This theorem is due to Bernstein, Khovanskii and MacPherson for the underlying complexes with rational coefficients. As a corollary the Euler characteristic Hodge numbers of non-degenerate toric hypersurface can be determined by the Euler characteristic subtotal Hodge numbers together with combinatorial data of Newton polyhedra. This is used implicitly in an explicit formula by Batyrev--Borisov. Note that a formula for the Euler characteristic subtotal Hodge numbers in terms of Newton polyhedra has been given by Danilov--Khovanskii. The structure theorem also implies that the graded quotients of the weight filtration on the middle cohomology of the canonical compactification of a non-degenerate toric hypersurface have Hodge level strictly smaller than the general case except for the middle weight. This gives another proof of a formula for the frontier Hodge numbers of non-degenerate toric hypersurfaces due to Danilov--Khovanskii.