# Which rings are cohomology rings?

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.

• A necessary condition is Poincaré duality. Perhaps the odd-degree cohomology groups have to be even-dimensional? (True in characteristic $0$.) – TKe Oct 8 '18 at 3:37
• Hard Lefschetz is also a necessary condition (for smooth projective varieties in any characteristic). – ulrich Oct 8 '18 at 4:36
• 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). – TKe Oct 8 '18 at 4:55
• 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. – Dan Petersen Oct 8 '18 at 5:03
• 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. – Donu Arapura Oct 8 '18 at 11:52