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Which rings can arise as cohomology rings of algebraic varieties?

To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth projective varieties over an algebraically closed field $k$.

Are there any conjectures and results giving criteria for a finite-dimensional graded-commutative $K$-algebra to be realizable as $H^*(V_k, K)$ for a smooth projective variety $V_k$ over $k$? Fixing such a $K$-algebra $A_K$ what can one say about the set of $V_k$ with $H^*(V_k, K)=A_K$?

I suppose the title question makes sense in purely topological settings too, for any cohomology theory with ring structure.

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    $\begingroup$ A necessary condition is Poincaré duality. Perhaps the odd-degree cohomology groups have to be even-dimensional? (True in characteristic $0$.) $\endgroup$
    – user19475
    Oct 8, 2018 at 3:37
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    $\begingroup$ Hard Lefschetz is also a necessary condition (for smooth projective varieties in any characteristic). $\endgroup$
    – naf
    Oct 8, 2018 at 4:36
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    $\begingroup$ Explicitly, hard Lefschetz implies that the odd and even degree $H$-Betti numbers $b_{2i}$ and $b_{2j+1}$ are non-decreasing for $2i,2j+1 \leq \mathrm{dim}\,X$ and also the primitive decomposition ([Kleiman, Algebraic cycles and the Weil conjectures], Proposition 1.4.2). $\endgroup$
    – user19475
    Oct 8, 2018 at 4:55
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    $\begingroup$ In the purely topological setting any $\mathbf Q$-algebra may be realized as the cohomology ring of a CW complex. This is a very special case of Sullivan's approach to rational homotopy theory. $\endgroup$
    – Dan Petersen
    Oct 8, 2018 at 5:03
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    $\begingroup$ There are several natural constraints coming from hard Lefschetz etc., as others have said. The hard direction seems to be the other way. While this doesn't answer it, you might find, Schreieder, On the construction problem for Hodge numbers useful to look at. $\endgroup$
    – Donu Arapura
    Oct 8, 2018 at 11:52

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