# Grothendieck's letter to Faltings: On the yoga of motives and the degeneration of Leray spectral sequence

In Grothendieck's letter to Faltings, he writes

There exists at this time a kind of “yoga des motifs”, which is familiar to a handful of initiates, and in some situations provides a ﬁrm support for guessing precise relations, which can then sometimes be actually proved in one way or another (somewhat as, in your last work, the statement on the Galois action on the Tate module of abelian varieties). It has the status, it seems to me, of some sort of secret science – Deligne seems to me to be the person who is most ﬂuent in it. His ﬁrst [published] work, about the degeneration of the Leray spectral sequence for a smooth proper map between algebraic varieties over $\mathbf C$, sprang from a simple reﬂection on “weights” of cohomology groups, which at that time was purely heuristic, but now (since the proof of the Weil conjectures) can be realised over an arbitrary base scheme.

Just what is this "simple reflection on 'weights' of cohomology groups" that he's referring to?? The proof in Deligne's paper doesn't use weights at all, but the relative version of the hard Lefschetz theorem.

Often when people say that a certain spectral sequence degenerates because of a weight argument, they mean that all differentials past a certain page go between cohomology groups of different weights. But this is not the case here: we can certainly today put a mixed Hodge structure and a weight filtration on all the terms in the sequence $$H^p(Y,R^qf_\ast \mathbf Q) \implies H^{p+q}(X,\mathbf Q)$$ (using Saito's theory of mixed Hodge modules), and if $f$ is smooth and proper then $R^qf_\ast \mathbf Q$ is a VHS pure of weight $q$. Even then, this does not imply anything about the weight filtration on $H^p(Y,R^qf_\ast \mathbf Q)$ unless we put more assumptions on $Y$ (and in Deligne's result $Y$ is arbitrary). If we in addition assume $Y$ to be smooth and proper then $E_2^{pq}$ is pure of weight $p+q$ and the spectral sequence will indeed degenerate for formal weight reasons, but surely this is not what Grothendieck is referring to?

• "... but surely this is not what Grothendieck is referring to" I think it is ! (although Grothendieck probably thought of Deligne's pure l-adic sheaves, not variations of Hodge structures). – Damian Rössler Aug 14 '13 at 8:30

Well, everything follows from the fact that the total direct image $Rf_*\mathbb{Q}$ splits as the direct sum of its (shifted, co)homology sheaves (so, it is "formal"). Now, this statement can be proved using weights of mixed Hodge modules; it is a consequence of the triviality of $DHM(Y)$-1-extensions between (pure) Hodge modules of the same weight (here I am using the convention in which $[q]$ increases weights by $q$).