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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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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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.

Théorie de HodgeIII, Pub. Math. IHES 44 (1974), p. 5-77. $\endgroup$1more comment