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 Xh. 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.
Algebraic foliations and derived geometry I: the Riemann-Hilbert correspondence / Gabriele Vezzosi. - In: SELECTA MATHEMATICA. NEW SERIES. - ISSN 1420-9020. - STAMPA. - ...:(In corso di stampa), pp. 1-44. [10.1007/s00029-022-00808-9]
Algebraic foliations and derived geometry I: the Riemann-Hilbert correspondence
Gabriele Vezzosi
In corso di stampa
Abstract
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 Xh. 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.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.