Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:28:37Z http://mathoverflow.net/feeds/question/106501 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106501/leray-degeneration-for-smooth-projective-morphisms-and-formality-of-families-of-c Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds Dan Petersen 2012-09-06T11:20:50Z 2012-09-07T21:09:37Z <p>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. </p> <p><strong>Question:</strong> Can this construction be made compatible with cup-product? That is, can one choose these injections so that the diagram $$\begin{matrix} R^i\pi_\ast\Q[-i] \otimes R^j\pi_\ast\Q[-j] &amp; \to &amp; R^{i+j}\pi_\ast\Q[-i-j] \\ \downarrow &amp; &amp; \downarrow \\ R\pi_\ast\Q \otimes R\pi_\ast\Q &amp; \to &amp; R\pi_\ast\Q \end{matrix}$$ commutes? If not, can one write down an obstruction?</p> <p>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 <em>algebras</em>. Hence an affirmative answer would be a version of DGMS's result which is moreover valid in families.</p>