  Let X be a complete complex nonsingular algebraic variety and D a closed
algebraic subset of X. We show that the cup product maps H^i(X \setminus D)
\otimes H^j(X,D) \to H^{i+j}(X,D) are morphisms of mixed Hodge structures. As a
corollary we obtain a new proof of a duality result of Fujiki. We also deal
with the extraordinary cup product.
