TITLE: Mixed Hodge structure and cup product
QUESTION [6 upvotes]: 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?

REPLY [9 votes]: 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.