Simpson's standard motivicity conjecture for connections

Let $X$ be a smooth complex projective variety, and let $n$ be a positive integer. The spaces $\mathcal{M}_{\mathrm{dR}}(X,n)$ and $\mathcal{M}_{\mathrm{B}}(X,n)$ denote respectively the moduli space of integrable connections of rank $n$ on $X$ and the moduli space of complex local systems of rank $n$ on $X$. An integrable connection is motivic if, after spreading out, it is a direct summand of a Gauss–Manin connection arising from a smooth projective family.

Simpson's standard conjecture. By spreading out, the $\overline{\mathbb{Q}}$-points in the intersection

$$\mathcal{M}_{\mathrm{B}}(X,n)(\overline{\mathbb{Q}})\bigcap \mathcal{M}_{\mathrm{dR}}(X,n)(\overline{\mathbb{Q}})$$

are motivic.

The conjecture addresses the transcendental nature of the Riemann–Hilbert correspondence and predicts that connections and monodromy representations simultaneously defined over $\overline{\mathbb{Q}}$ originate from geometry. Its relationship with rigidity motivates the weaker conjecture that rigid integrable connections are motivic.