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I need a hint or a good reference for definition of mixed Hodge structure on the relative cohomology groups ($\mathrm{H}^*(X,Y)$, $Y\subset X$ a closed subvariety of a comolex quasiprojective variety $X$, or even(?) $Y$ is a cycle (formal sum of subvarities) in $X$ ).

Thanks!

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  • $\begingroup$ This is even easier than the complement: just consider the kernel of the restriction map $\Omega^*(X)\to\Omega^*(Y)$; it computes what you want and, by Deligne, it has a mixed Hodge structure. $\endgroup$
    – Alex Degtyarev
    Commented Feb 18, 2015 at 20:32
  • $\begingroup$ @AlexDegtyarev You mean to consider the Hodge complex structure on the kernel? Does there exist a simpler way? $\endgroup$
    – Mostafa - Free Palestine
    Commented Feb 18, 2015 at 20:43
  • $\begingroup$ What can possibly be simpler? You don't even need logarithmic differentials! And, of course, this is all described in Deligne's three papers very nicely. $\endgroup$
    – Alex Degtyarev
    Commented Feb 18, 2015 at 21:23

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I suppose that the reference is Deligne, Théorie de Hodge III, 8.3.8. Also see section 5.5 of Peters and Steenbrink's book on Mixed Hodge structures.

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