# Simpson's motivicity conjecture

My question is about Simpson's motivicity conjecture, that is the conjecture that for any (cohomogically) rigid irreducible connection $$(M,\nabla)$$ on a smooth complex scheme $$X$$ is of geometric origin in the sense that there exists $$Y\overset{f}{\to}X$$ such that $$(M,\nabla)$$ is a subquotient of $$R^nf_*\mathcal{O}_Y$$ with the Gauss-Manin connection.

My main question can be put over the top as "why should we believe it?" I'm aware that predictions of this conjecture have been proved, for instance by H.Ésnault and her colaborators. I'd be more interested in a "plausability" sort of criterion. For instance, there are some deformation theoretic arguments explaining why the Fontaine-Mazur conjecture should hold (I'm choosing F-M because it is of a very similar "shape"). Considering this came from Simpson's work on non-abelian Hodge theory, presumably there is a non-abelian Hodge theoretic reason to expect the validity of this conjecture...

Also, how "sharp" do we expect this conjecture to be? Do we know that subquotients of Gauss-Manin connections are rigid?

At the time Simpson formulated his conjecture, he had proved that rigid local systems correspond to rational (in a suitable sense) variations of Hodge structure. And, as you point out, we now know that they are integral (Esnault-Groechenig). So rigid local systems have many of the earmarks of geometric local systems.

As for your other questions, geometric local systems, or their subquotients (which would be rationally summands), need not be rigid.

• Thanks! Is there a "good" reason why we think we can characterise (conjecturally) the geometric Galois representations but not those of (topological) fundamental groups? Mar 25 at 17:59
• @curiousmathguy I don't have a good answer, except to observe that the structures arising from $p$-adic Hodge theory are very rich, and perhaps that's lacking to some extent on the complex side. Mar 25 at 18:08
• @curiousmathguy Galois groups are so rich and complicated that it's sort of a miracle any representations with large image exist at all. Topological fundamental groups can be presented by generators and relations, so a representation is just a solution of some system of polynomial equations, which often come in large deformation spaces, and it's harder to characterize the sparse points that arise from geometry among them. Mar 26 at 15:55

I'm not sure if this is the kind of evidence you are looking for, but since you mention the Fontaine-Mazur conjecture, let me remark that the relative version of the Fontaine-Mazur conjecture implies Simpson's motivicity conjecture.

A rigid irreducible local system, in particular, is arithmetic: it extends to an etale local system on the descent of your complex variety to some finitely generated field. This is Theorem 4 in Simpson's http://www.numdam.org/article/PMIHES_1992__75__5_0.pdf (note that every local system of geometric origin is arithmetic, by a spreading out argument).

Now, one can show that for any irreducible arithmetic local system, the extension to an etale local system over a finitely generated field can be chosen to satisfy the assumptions of the relative Fontaine-Mazur conjecture (e.g. as formulated by Liu and Zhu here https://arxiv.org/abs/1602.06282) -- this follows from the main result of https://arxiv.org/abs/2012.13372 See Lemma 6.2 there for this particular statement.

This deduction of Simpson's conjecture from the Fontaine-Mazur conjecture is also outlined on the last page of the paper https://arxiv.org/abs/2101.00487 by Esnault and Kerz.