Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection $$ \newcommand{\Q}{\mathbf{Q}} R^i \pi_\ast\Q[-i] \hookrightarrow R\pi_\ast\Q,$$ such that the direct sum of all these gives a quasi-isomorphism between $R\pi_\ast\Q$ and its cohomology.

Question: Can this construction be made compatible with cup-product? That is, can one choose these injections so that the diagram $$ \begin{matrix} R^i\pi_\ast\Q[-i] \otimes R^j\pi_\ast\Q[-j] & \to & R^{i+j}\pi_\ast\Q[-i-j] \\\\ \downarrow & & \downarrow \\\\ R\pi_\ast\Q \otimes R\pi_\ast\Q & \to & R\pi_\ast\Q \end{matrix}$$ commutes? If not, can one write down an obstruction?

The question is motivated by the fact that the fibers are compact Kähler manifolds, hence formal by Deligne-Griffiths-Morgan-Sullivan. So on each fiber $X_s$, we have a quasi-isomorphism with the cohomology when both are considered as dg algebras. Hence an affirmative answer would be a version of DGMS's result which is moreover valid in families.

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    In general, it is not true that this construction can be made compatible with cup products. I had thought about this question many years ago and, if I remember correctly, there are obstructions even when the relative dimension of $\pi$ is one. See the recent paper of Voisin "Chow rings and decomposition theorems..." for more information. – ulrich Sep 8 '12 at 6:43
  • @ulrich Thanks for the reference! Voisin's paper is really interesting and settles the question authoritatively. If you re-post your comment as an answer I'll accept it. – Dan Petersen Sep 8 '12 at 13:51

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