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Let $f\colon X\to \mathbb{A}^1$ be a smooth projective morphism of complex algebraic manifolds, where the target $\mathbb{A}^1$ is the affine line. Are there any restrictions on the Hodge structures ...

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?...

Apologies if the question in the title is too elementary. A reference would be helpful.

What is the first work that studies, refers to, or mentions the Hodge characteristic? The Hodge polynomial is the unique ring homomorphism $$ P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{...