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.
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44
Gabriele Vezzosi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2158/1286108
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