Let $X$ be a smooth projective variety over $\mathbb{C}$ with dimension $n$. Is it true that for every $i<n$, the Hodge structure on $\mathrm{H}^i(X,\mathbb{Q})$ is generated by Hodge structures of cohomology groups of varieties of lower dimension? (i.e. subquotient of direct sums)
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asked Jan 28, 2015 at 21:11
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2$\begingroup$ Not for $n=2$ and $i=1$; take $X$ the variety of lines on a smooth cubic threefold, then this is basically the main result of the Clemens-Griffiths proof of irrationality of smooth cubic threefolds. My guess is that for $i$ at least $2$ this is a hard question. $\endgroup$– dhyCommented Jan 28, 2015 at 21:28
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2$\begingroup$ This will follow from Lefschetz hyperplane section theorem $\endgroup$– VenkataramanaCommented Jan 29, 2015 at 6:05
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$\begingroup$ I am not sure I understand the question. Let us take $n=2$, $i=1$, so $X$ is a surface; what do you mean by "the Hodge structure on $H^1(X,\mathbb{Q})$ is generated by Hodge structures of curves"? $\endgroup$– abxCommented Jan 29, 2015 at 7:57
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$\begingroup$ @abx I mean it is a subquotient of a direct sum of Hodge structures of cohomology groups of curves. I think lefschetz hyperplane implies the above claim for $i< n-1$ but I am curious about the case $i=n-1$ $\endgroup$– Mostafa - Free PalestineCommented Jan 29, 2015 at 8:10
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1$\begingroup$ So, $H^{n-1}(X,Q)$ is a substructure in the corresponding cohomology of a smooth hyperplane section of $X$. Now you should just recall that the category of polarizable (pure) Hodge structures is Abelian semi-simple. $\endgroup$– Mikhail BondarkoCommented Jan 29, 2015 at 8:13
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