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Let $X$ be a smooth projective complex algebraic variety. The Hodge decomposition tells us that $H^n(X, \mathbf C) = \oplus H^{p,q}$.

Here is my question:

For what kind of $X$ is $H^{2n}(X) = H^{n,n}$ for all $n$ (I don't care about the odd cohomology groups $H^{2n+1}$)? Has this condition been studied in the literature?

Or a slightly more relaxed question: given some $X$, can one construct a related variety $X'$ so that the property holds for $X'$? Or, maybe, give bounds on $dim H^{2n} - dim H^{n,n}$ etc.

Thank you!

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    $\begingroup$ This condition holds whenever the derived category of coherent sheaves has an exceptional collection(odd H^* will be zero in this case). If you take a variety with an exceptional collection and blow it up at a point and it will still have this property. $\endgroup$
    – Daniel Pomerleano
    Commented Oct 9, 2014 at 14:03
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    $\begingroup$ See the question mathoverflow.net/questions/151341 $\endgroup$
    – Dan Petersen
    Commented Oct 9, 2014 at 14:05
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    $\begingroup$ Cellular varieties have this property. But not only these. As for constructing a "related variety $X'$", I think there is no chance. What would you expect to get for a curve of positive genus? $\endgroup$
    – Sasha
    Commented Oct 9, 2014 at 16:30
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    $\begingroup$ As others say, it's very natural to ask that all cohomology be in $H^{p,p}$, and therefore that the odd cohomology vanishes; I think it very strange to ask this only of the even cohomology. $\endgroup$
    – Allen Knutson
    Commented Oct 9, 2014 at 17:58
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    $\begingroup$ One example where the even cohomology is spanned by algebraic cycles and the odd cohomology is nonzero is the moduli space $\overline M_{1,n}$ of $n$-pointed stable curves of genus one, which has odd cohomology for $n\geq 11$. $\endgroup$
    – Dan Petersen
    Commented Oct 10, 2014 at 8:06

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A more restrictive condition, which probably has been studied, is to look at varieties with Hodge level $\le 1$. This means that $h^{pq}=0$ unless $|p-q|\le 1$. If you assume the general Hodge conjecture, then this is equivalent to the motive being a summand of $\bigoplus C_i(n_i)$ for some smooth projective curves $C_i$. If just want examples, they are easy enough to find. Take your favourite variety with cohomology generated by algebraic cycles (e.g. a toric variety, a flag variety…) and then optionally take the product with a curve.

I am a bit skeptical about the last two questions. If $X$ is a degree $d$ surface, then $\dim H^2 - h^{11}$ grows with $d$. I am not sure what you would take for $X'$ in such a case.

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