# Mixed Hodge structure and cup product

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?

• Is the variety over $\mathbb{C}$? Mar 16 '15 at 10:46
• 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.
– abx
Mar 16 '15 at 11:11
• @abx You should write that as an answer! Mar 16 '15 at 14:25
• 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? Mar 16 '15 at 14:26
• 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?) Mar 16 '15 at 15:41