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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.

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  • $\begingroup$ In the complex case, the Leray spectral sequence degenerates at $E_2$ when the family $f \colon X \to Y$ is proper and smooth (i.e., all fibres are smooth). This was proven by Deligne. Maybe does his proof also work somehow in your context? $\endgroup$ Feb 4, 2014 at 14:00
  • $\begingroup$ @FrancescoPolizzi — Thanks for your comment. What exactly do you mean with “degenerates”? (It usually only means the filtration, not the splitting.) Anyway, do you have a reference for Deligne's proof? $\endgroup$
    – jmc
    Feb 4, 2014 at 14:05
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    $\begingroup$ It means that all the differentials are zero at the $E_2$ level, so you have your (non-canonical) decomposition. A reference is the paper by De Cataldo and Migliorini The Decomposition Theorem, perverse sheaves and the topology of algebraic maps, see in particular Theorem 1.5 (Deligne's papers are quoted just above it). google.it/… $\endgroup$ Feb 4, 2014 at 14:10
  • $\begingroup$ I suppose you are actually looking at $H^*(\bar{X},\mathbb{Q}_l)$, with $\bar{X}=X\times _k\bar{k}$ -- otherwise you have no Galois action. $\endgroup$
    – abx
    Feb 4, 2014 at 14:40
  • $\begingroup$ @abx — You're absolutely right. I'll edit it in. $\endgroup$
    – jmc
    Feb 4, 2014 at 14:42

1 Answer 1

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

The important references for this are

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.
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  • $\begingroup$ Hi Johan, I agree with you that the decomposition is Galois equivariant. You are also right that there is a more general version of your answer using the decomposition theorem. The point is that there is a funny "perverse" t-structure on $D^b(\overline Y)$ for which $Rf_\ast \mathbf Q_\ell \cong \oplus_i H^i(Rf_\ast\mathbf Q_\ell)[-i]$ holds for arbitrary proper morphisms $f$, here $H^i(-)$ means cohomology sheaves with respect to this perverse t-structure. This is one way of formulating the decomposition theorem. If $X$ didn't happen to be smooth we should replace $\mathbf Q_\ell$ ... $\endgroup$ Feb 5, 2014 at 22:03
  • $\begingroup$ with the intersection complex $IC_X$. In any case the decomposition theorem does not imply that the filtration for the ordinary Leray spectral sequence splits in this generality, only for the perverse Leray spectral sequence. What you can say is that if all the perverse sheaves $H^i(Rf_\ast \mathbf Q_\ell)$ occuring in the decomposition happen to be actual sheaves (as opposed to complexes of sheaves), then the Leray spectral sequence degenerates and the filtration splits. $\endgroup$ Feb 5, 2014 at 22:03
  • $\begingroup$ (Here's some self-promotion: in Section 2 of my preprint arxiv.org/abs/1310.7369 is an example of proving Leray degeneration for a specific proper but not smooth morphism by precisely this method: checking that the perverse sheaves in the derived pushforward are actual sheaves.) $\endgroup$ Feb 5, 2014 at 22:08
  • $\begingroup$ @DanPetersen — Thanks for the extensive comments! I guess I am going to study the perverse situation pretty soon. (But I do not know anything yet about t-structures, &c.) First I want a better understanding of the smooth projective situation. I will definitely look at your preprint. Thanks again! $\endgroup$
    – jmc
    Feb 6, 2014 at 7:09

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