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Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).

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  • $\begingroup$ Grothendieck's original motivation was to find a Tannakian category of motives. If you have a Tannakian category, and a fiber functor on that category, then you can construct an algebraic group from its automorphisms (Deligne's reconstruction theorem). Probably the reason these are related are the many incarnations of the Galois group (or fundamental group, or something else) and their actions on appropriate cohomology groups. It's only a step from seeing these groups act on the cohomology to trying to functorialize and find the root of where the two come from. $\endgroup$
    – Eoin
    Commented Aug 16, 2017 at 16:28
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    $\begingroup$ I'm quite sketchy on this so please correct me if I'm wrong. There are many realisations of a motive, including Betti and de Rham realisations, which together with the Betti-de Rham comparison isomorphism form the "Hodge realisation". There are also $l$-adic realisations for prime numbers $l$. Conjecturally, the Hodge and $l$-adic realisations should both be fully faithful functors. Thus the Betti fibre functor, and the Tannaka group it defines, is a good approximation to the full "motivic" Galois group, and the extra info provided by the Hodge realisation should capture everything motivic. $\endgroup$
    – Alex Saad
    Commented Aug 16, 2017 at 16:44
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    $\begingroup$ @AlexSaad There are actually two different conjectural faithful functors related to the Hodge-de Rham isomorphism. The first is based on the $\mathbb C$-de Rham cohomology w/ its Hodge filtration and the comparison isomorphism, and this faithfulness is the Hodge conjecture. The second is based on the $\mathbb Q$-de Rham cohomology and the comparison isomorphism after tensoring with $\mathbb C$, and this faithfulness is the period conjecture. $\endgroup$
    – Will Sawin
    Commented Aug 16, 2017 at 17:37
  • $\begingroup$ Just recalling that usually the motivic t-strucure is defined in terms of the Betti realization when it exists. $\endgroup$
    – user40276
    Commented Aug 22, 2017 at 20:46

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