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I'm looking for a reference for the answer to the following questions.

Let $X$ be an algebraic variety over C. When is the cup product a morphism of Mixed Hodge structures? Does $X$ have to be smooth?

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  • $\begingroup$ Is the variety over $\mathbb{C}$? $\endgroup$ Mar 16 '15 at 10:46
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    $\begingroup$ This is true with no hypothesis on $X$: see Corollaire 8.2.11 in Deligne Théorie de Hodge III, Pub. Math. IHES 44 (1974), p. 5-77. $\endgroup$
    – abx
    Mar 16 '15 at 11:11
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    $\begingroup$ @abx You should write that as an answer! $\endgroup$
    – Ben Webster
    Mar 16 '15 at 14:25
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    $\begingroup$ Also, seriously? 3 close votes? I understand it's not the best written question and the answer is reasonably "well-known," but mixed Hodge structures are too basic for this site now? $\endgroup$
    – Ben Webster
    Mar 16 '15 at 14:26
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    $\begingroup$ I had similar thoughts. I wonder how many people who voted to close knew the answer. (Test: Is the category of mixed Hodge structures Tannakian, true of false?) $\endgroup$ Mar 16 '15 at 15:41
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This is true with no hypothesis on X: see Corollaire 8.2.11 in Deligne Théorie de Hodge III, Pub. Math. IHES 44 (1974), p. 5-77.

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