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Simpson's motivicity conjecture says that for any rigid, flat irreducible connection $(V,\nabla)$ on a smooth complex variety $M$, there exists a proper smooth morphism $f:X \to M$ s.t. $(V,\nabla)$ is a subquotient of the Gauss-Manin connection $R^if_*\mathcal{O}_X$.

Here is my question. Why does the above conjecture require the rigidity of the connection?

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    $\begingroup$ Your definition of the Gauss-Manin connection is not quite correct, one should look at the connection on relative de Rham cohomology, not the cohomology of the structure sheaf. $\endgroup$
    – naf
    Commented Nov 18, 2023 at 8:30
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    $\begingroup$ Rigidity is not an essential condition, but if one drops it then one must impose some other condition. One reason for this is that a subquotient of a Gauss-Manin system always has quasi-unipotent local monodromy (at infinity), but it is very easy to find irreducible connections which do not have this property. An example of another condition that one can impose is to require that the connection comes from a polarised variation of Hodge structure; Simpson also has a conjecture in this context. (Rigid connections are quite rare, most Gauss-Manin systems are not be rigid.) $\endgroup$
    – naf
    Commented Nov 18, 2023 at 8:37

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In the 90's, Simpson proved that rigid local systems on a projective variety come from complex variations of Hodge structure, so it seemed a natural if bold leap to conjecture that they come from geometry. Recently, Esnault and Groechenig have proved that (cohomologically) rigid local systems are integral. This gives further evidence.

If you drop rigidity, then most local systems are not motivic. This is clear already for rank one local systems on an elliptic curve. A reasonable conjecture might be that any local system can be deformed to a motivic local system.

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  • $\begingroup$ Could you say what the obstruction is for local systems on an elliptic curve? $\endgroup$
    – user19232801
    Commented Nov 18, 2023 at 13:49
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    $\begingroup$ Motivic local systems should be rational i.e. they should lie in $Hom(\mathbb{Z}^2,\mathbb{Q}^*)$, but clearly $Hom(\mathbb{Z}^2,\mathbb{C}^*)$ is bigger. $\endgroup$
    – Donu Arapura
    Commented Nov 18, 2023 at 13:55
  • $\begingroup$ I'm probably misunderstanding the definitions (or maybe the OP stated them wrong), but why can't you get algebraic integers? E.g. local systems with third roots of unity arise as summands of the pushforward of a triple cover of the curve. $\endgroup$
    – user19232801
    Commented Nov 18, 2023 at 20:23
  • $\begingroup$ @user19232801 The term motivic local system is a bit ambiguous. For me, it would be mean that the local system is defined over $\mathbb{Q}$ among other things.... $\endgroup$
    – Donu Arapura
    Commented Nov 20, 2023 at 16:00

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