When does the filtration in the limit of the Leray spectral sequence split?

Question

Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says $$ E_{2}^{pq} = H^{p}(\bar{Y},R^{q}f_{*}\mathbb{Q}_{\ell}) \ \Longrightarrow\ H^{p+q}(\bar{X}, \mathbb{Q}_{\ell}) . $$ In particular this means that there is a filtration on $H^{n}(\bar{X},\mathbb{Q}_{\ell})$ such that $E_{\infty}^{pq}$ (with $p+q = n$) are the graded pieces.

Question: Are there conditions on $f$, $X$, and/or $Y$ that make this filtration split?

(This means that $H^{n}(\bar{X},\mathbb{Q}_{\ell}) \cong \bigoplus_{p+q=n} E_{\infty}^{pq}$; even if the direct sum decomposition is not canonical.)

Let me point out explicitly that I am looking for a decomposition that is Galois equivariant.

Answer 1

Credits to Francesco Polizzi for putting me on the right track, by pointing to Deligne's work.

The important references for this are

• Deligne's thesis “Théorème de Lefschetz et critères de dégénérescence de suites spectrales” Publications Mathématiques de l'IHÉS. 35 (1968) pp. 107–126.
• Deligne's paper “Décompositions dans la catégorie dérivée” in: Motives. Proceedings of Symposia in Pure Mathematics. 55 t1 (AMS 1994) pp. 115–128.

I guess if one invokes perverse sheaves there might be more general answers. If anyone would like to give such a more general answer, I would be very grateful!

[Edit] As Dan points out below, working with perverse sheaves indeed helps: then there is a decomposition if $f$ is proper. [/Edit]

What follows requires that $f$ is smooth (and proper, which was already an assumption in the question). Here goes my attempt:

The decomposition

In Deligne's thesis “Théorème de Lefschetz et critères de dégénérescence de suites spectrales” we find the following results:

• Proposition 2.1 states that in the derived category $D^{\textrm{b}}(\bar{Y})$ we have an isomorphism $$ \textrm{R}f_{*} \mathbb{Q}_{\ell} \cong \bigoplus_{i} \textrm{R}^{i}f_{*} \mathbb{Q}_{\ell}[-i]. $$ The proposition applies, since $\mathbb{Q}_{\ell}$ satisfies the Lefschetz condition by the relative hard Lefschetz theorem (this uses that $f$ is smooth and proper).
• Proposition 2.16 with $Z = \textrm{Spec}(k)$, can be applied (since $Y$ is smooth over $k$; and by the previous point). This gives $$ \textrm{R}(\Gamma f)_{*} \mathbb{Q}_{\ell} \cong \bigoplus_{k} \textrm{R}\Gamma_{*} (\textrm{R}f_{*}^{k} \mathbb{Q}_{\ell}[-k]). $$

Taking the $n$-th homology group, then gives $$ \textrm{H}^{n}(\bar{X}, \mathbb{Q}_{\ell}) = \bigoplus_{q} \textrm{H}^{n}(\bar{Y}, \textrm{R}f_{*}^{q} \mathbb{Q}_{\ell}[-q]) = \bigoplus_{q} \textrm{H}^{n-q}(\bar{Y}, \textrm{R}f_{*}^{q} \mathbb{Q}_{\ell}). $$ This last expression can of course be written in the usual way: $$ \textrm{H}^{n}(\bar{X}, \mathbb{Q}_{\ell}) = \bigoplus_{p+q=n} \textrm{H}^{p}(\bar{Y}, \textrm{R}f_{*}^{q} \mathbb{Q}_{\ell}). $$

The Galois equivariance

If I am not mistaken, the above decomposition is Galois equivariant. I hope some expert can point out errors if I go astray. Here it goes:

• Deligne starts with a very general Proposition (1.2) where he works with the bounded derived category $D^{\textrm{b}}(A)$ of some abelian category $A$. In particular we can work with the category $D^{\textrm{b}}(Y \times_{k} \bar{k})$ of lisse $\mathbb{Q}_{\ell}$-sheaves together with the Galois action. Here one has to identify $\sigma^{*}\mathcal{F}$ with $\mathcal{F}$, just as when one defines the Galois action on $\ell$-adic cohomology (or when proving the functoriality of $\textrm{H}_{\textrm{ét}}^{i}(\_, \mathbb{Q}_{\ell})$).
• The next Theorem we need is (1.5), and this is Galois equivariant because the hard Lefschetz theorem is Galois equivariant. (Here I bluf that this also holds for the relative hard Lefschetz theorem.)
• Consequently the decomposition of Proposition 2.1 is Galois equivariant.
• And finally, therefore the decomposition of Proposition 2.16 is also Galois equivariant.