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There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge decomposition, etc.

Is anything known in any kind of generality about the torsion in the singular cohomology beyond the results which hold for arbitrary compact (smooth) manifolds (i.e. Poincaré duality)? I'm primarily interested in hypersurfaces in $\mathbb{P}^n$, so I'd love to hear results that are true in this case (or for complete intersections, etc.), even if they require further restriction to low dimension or something.

The only result I know of which mentions the integral singular cohomology is the Lefschetz hyperplane theorem, and this doesn't obviously (to me at least!) imply anything special about what the torsion can look like.

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There isn't any torsion in the cohomology of (smooth) hypersurfaces in $\mathbb{CP}^n$. This follows from the Lefschetz hyperplane theorem together with the universal coefficient theorem; see this blog post for details. I think the same is true for (smooth) complete intersections.

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  • $\begingroup$ Wow - this is way better than anything I hoped for! I suppose the non-smooth case is much harder? $\endgroup$
    – dorebell
    Aug 13 '15 at 0:36

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