Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety $X$ is formally integrable at any point, and, if we suppose that its singular locus has codimension $\geq 2$, then the truncation of its analytification is a locally integrable singular foliation on the associated complex manifold $X^h$. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on $X$, and prove a Riemann-Hilbert correspondence for them when $X$ is proper. We discuss several examples and applications.