Hodge theoretic properties of intersection cohomology

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?

I am not an expert in the field.

• 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".
– dhy
Jan 2, 2021 at 18:07
• @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 $i$.
– asv
Jan 2, 2021 at 18:16
• 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.
– dhy
Jan 2, 2021 at 18:25

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.

• Thank you. To make sure: are there Hodge-Riemann relations? Do you have a reference to them?
– asv
Jan 3, 2021 at 6:28
• 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. Jan 3, 2021 at 7:29
• 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. Jan 3, 2021 at 7:30