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Let $X$ be a complex projective irreducible reduced variety. It is well known that the intersection cohomology of $X$ satisfies versions of Poincare duality and hard Lefschetz theorem.

Does it admit a pure Hodge structure? If yes, does the latter satisfy the Hodge-Riemann bilinear relations?

A reference would be helpful.

I am not an expert in the field.

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    $\begingroup$ My understanding (but it's been a long time since I've thought about any of this): It does admit a pure Hodge structure, but there is no good cup product on intersection cohomology, so I'm not sure how to make sense of "satisfies the Hodge-Riemann bilinear relations". $\endgroup$
    – dhy
    Jan 2, 2021 at 18:07
  • $\begingroup$ @dhy: My guess is that for that one may use the Poincare pairing. For smooth varieties one uses for the latter the cup product, put in singular case it is avoided. I think this pairing still can be used to pair $[\omega]^{n-i}x$ with $\bar x$ where $x$ is an intersection coholology class of degree $i$. $\endgroup$
    – asv
    Jan 2, 2021 at 18:16
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    $\begingroup$ Ah, I see. In that case, I think the answer is yes, they do satisfy the Hodge-Riemann bilinear relations. I learned about this from de Cataldo-Migliorini's "The decomposition theorem, perverse sheaves and the topology of algebraic maps", but they cite some papers of Saito as the original source. $\endgroup$
    – dhy
    Jan 2, 2021 at 18:25

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As dhy suggested in a comment, this is indeed true and it is part of what Saito proved in his formalism of mixed Hodge modules. A mixed Hodge module is essentially a perverse sheaf carrying something like a variation of mixed Hodge structure. Saito constructed a six functor formalism on the derived category of mixed Hodge modules, and enhanced the intersection complex to a mixed Hodge module, in particular putting a (pure) Hodge structure on intersection cohomology.

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  • $\begingroup$ Thank you. To make sure: are there Hodge-Riemann relations? Do you have a reference to them? $\endgroup$
    – asv
    Jan 3, 2021 at 6:28
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    $\begingroup$ Yes, it's true, and I should have said this. A mixed Hodge module is rather something like a variation of polarizable mixed Hodge structure, and a mixed Hodge module on a point is a graded-polarizable mixed Hodge structure, so this is the structure you get on intersection cohomology. I'm not sure off hand whether he proved the precise statement that an ample class defines a polarization but if not I'd bet it can be extracted from what he proved. Original references are "Modules de Hodge polarisables" and "Mixed Hodge modules", both in Publ. RIMS. $\endgroup$ Jan 3, 2021 at 7:29
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    $\begingroup$ A useful executive summary is in a brief note, "Introduction to mixed Hodge modules". I'd prefer not to dig through Saito's papers but let me know if you can't figure it out. $\endgroup$ Jan 3, 2021 at 7:30

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