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It is a classical fact that the cohomology ring (with complex coefficients) of a complex smooth projective manifold is a bigraded algebra satisfying (1) Poincare duality; (2) hard Lefschetz theorem; (3) Hodge-Riemann bilinear relations.

On the other hand in combinatorics there are examples of different origin of algebras with all these properties (see e.g. Combinatorial applications of Hodge-Riemann relations by J. Huh).

I am wondering whether it is possible to prove that at least some of these algebras coming from combinatorics are not isomorphic to cohomology ring of any manifold as above so that the isomorphism preserves all structures. A closely related question is whether there are extra properties of the cohomology ring of any smooth projective complex manifold.

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    $\begingroup$ Not exactly what you are asking, but I remember June Huh answering a question after a talk saying that there exist Lorentzian polynomials that are not volume polynomials of projective varieties. (You can hear him explain this around the 1:04 minute mark of youtube.com/watch?v=s9hKZ42HLfk) This means the combinatorial theory is more general than the geometric theory. Of course, I do not know if these Lorentzian polynomials have any particular combinatorial significance. $\endgroup$
    – Sam Hopkins
    Commented Jan 10, 2023 at 13:45

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