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Deligne has a theorem in "Theorie de Hodge II" as follows:

Let $S$ be a smooth separated scheme, and $f:X\to S$ be a smooth proper morphism. Let $\bar{X}$ be a non singular compactification of $X$. Then the canonical morphism : $$ H^n(\bar{X},\mathbf{Q)}\to H^0(S,\mathbf{R}^nf_* \mathbf{Q}) $$ is surjective.

My question is: if we replace the constant sheaf $\bf Q$ over $\bar{X}$ with a local system $E$ (a locally constant sheaf) , does the theorem still hold ?

Thank you !

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1 Answer 1

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Yes, this is still true, if we assume that $E$ is the underlying local system of a polarized variation of Hodge structure on $\overline X$, which takes care of most local system of "algebro-geometric origin".

Deligne's result comes from a combination of the following three results:

  1. The Leray spectral sequence for $f \colon X \to S$ and the sheaf $\mathbf Q$ degenerates.
  2. The map $H^n(\overline X,\mathbf Q) \to W_nH^n(X,\mathbf Q)$ is surjective.
  3. $H^0(S,R^nf_\ast \mathbf Q)$ is pure.

To generalize to a local system we need a suitable formalism of mixed sheaves. Since you have $\mathbf Q$-coefficients it looks like you're working over $\mathbf C$ so we should use Saito's theory of mixed Hodge modules. Then all these remain true with coefficients in $E$ instead.

For the first one, Saito proves that the perverse Leray sequence degenerates for a proper morphism and a pure Hodge module. For a morphism which is in addition smooth the perverse Leray sequence is just the ordinary one, and if $E$ is a PVHS then it's a pure Hodge module.

For the second it is enough to prove dually that $\mathrm{gr}^W_{n+k} H^n_c(X,E) \to \mathrm{gr}^W_{n+k} H^n_c(\overline X, E)$ is injective (where $k$ is the weight of the sheaf $E$). But the long exact sequence of a pair identifies the kernel of this map with a quotient of $\mathrm{gr}^W_{n+k} H^{n-1}_c(\overline X \setminus X,E)$ which vanishes by the fact that $Rf_!$ decreases weights. (For a more general statement see Peters and Saito, "Lowest weights in cohomology of variations of Hodge structure".)

The third follows because $H^0(S,R^nf_\ast E)$ injects into $H^n(X_s,E)$ (as the monodromy invariants) which is pure because $X_s$ is smooth and proper and because the restriction of $E$ to $X_s$ is still pure, again everything is due to Saito's theory.

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Can you give me more detail reference ? Or some body proved that theorem in a published paper? For we want to cite it. –  Lan Sep 9 '13 at 13:27
2  
I don't know a paper proving this specific fact, most likely you'll have to include it as a lemma with a proof along what I wrote above. Saito's papers are notoriously difficult to read, but two references which could be useful for you are the first few pages of Saito's "Introduction to mixed Hodge modules" (which should contain enough information to fill in the details in what I wrote) and the treatment in the book of Peters and Steenbrink. Or you could read Beilinson-Bernstein-Deligne (for the l-adic story), or Brylinski and Zucker's "An overview of recent advances in Hodge theory". –  Dan Petersen Sep 9 '13 at 16:10

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