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Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a quasicoherent $O_S$--module, carrying a canonical and natural integrable connection - the "Gauss-Manin connection" (Katz-Oda 1968). Its construction following Katz and Oda uses the spectral sequence associated with the higher direct image functors $R^i\pi_\ast$ and a canonical filtration of the differential complex $\Omega_{X/k}^\ast$.

Now suppose $M$ is a motive over $S$, say in the sense of Jannsen or a 1-motive as defined by Deligne, and denote by $T_{dR}(M/S)$ its de Rham realisation. This is a quasicoherent $O_S$--module which, according to the general philosophy, must come with a canonical and natural integrable connection.

How can we construct this "Gauss-Manin connection" on $T_{dR}(M/S)$?

Just walking through the Katz-Oda paper and replacing $X$'es by $M$'s will not do, because there are no such things as $\pi_\ast$ or $\Omega_{M/k}^\ast$.

In the case $k=\mathbb C$, we can use comparison of de Rham realisation with Betti realisation (the motivic instance of de Rham's theorem, which is forced to hold in Jannsen's setting, and which proven to hold for 1--motives by Deligne) and produce the sought connextion analytically, but that is not what I want.

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It seems that your question is really about Jannsen's definition of motive, so you might get a better response if you recall what that is. I'm certainly not familiar with it. But anyway, I would guess that a basic result in his theory would be that any "Weil" cohomology functor (i.e. one satisfying certain basic properties) on the category of smooth S-schemes would factor through his category of motives. Or if he defines his category using explicit realizations, then just take the de Rham component of the realization! But again, I'm just guessing. –  JBorger May 26 '10 at 12:36
    
Ok, just to give the idea: A mixed realisation data over a finitely generated field of char zero is a compatible system consisting of a Hodge structure for each complex embedding of $k$, a $p$--adic Galois representation for each prime number $p$ and a filtered $k$-vector space, together with comparison isomorphisms. These things form a Tannakian category. To each variety $V$ over $k$ one can associate a realisation datum $H(V)$ via various cohomology theories. Jannsen's category of mixed motives is the Tannakian subcategory generated by the image of $H$. –  Xandi Tuni May 26 '10 at 13:14
    
Okay, but what do you have in mind when you speak of a motive over $S$? I suspect that this is what James is getting at. –  Emerton May 29 '10 at 3:30

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