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 $\mathbb{C}$? $\endgroup$– Marco GollaCommented Mar 16, 2015 at 10:46
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5$\begingroup$ 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. $\endgroup$– abxCommented Mar 16, 2015 at 11:11
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2$\begingroup$ @abx You should write that as an answer! $\endgroup$– Ben Webster ♦Commented Mar 16, 2015 at 14:25
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9$\begingroup$ 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 ♦Commented Mar 16, 2015 at 14:26
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2$\begingroup$ 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?) $\endgroup$– Donu ArapuraCommented Mar 16, 2015 at 15:41
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1 Answer
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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.
