ALGEBRAIC FOLIATIONS AND DERIVED GEOMETRY: THE RIEMANN-HILBERT CORRESPONDENCE

Bertrand Toën (1) , Gabriele Vezzosi (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 DiMal - Dipartimento di Matematica e Informatica [Firenze]

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 ≥ 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.

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]

Bertrand Toën, Gabriele Vezzosi. ALGEBRAIC FOLIATIONS AND DERIVED GEOMETRY: THE RIEMANN-HILBERT CORRESPONDENCE. Selecta Mathematica (New Series), 2022, ⟨10.1007/s00029-022-00808-9⟩. ⟨hal-02441660⟩