Algebraic foliations and derived geometry: the Riemann-Hilbert
correspondence
B. Toën, and G. Vezzosi. (2020)cite arxiv:2001.05450Comment: Added a few results. Submitted version.
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 $\geq
2$, 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.
Description
Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence
%0 Generic
%1 toen2020algebraic
%A Toën, Bertrand
%A Vezzosi, Gabriele
%D 2020
%K Riemann-Hilbert
%T Algebraic foliations and derived geometry: the Riemann-Hilbert
correspondence
%U http://arxiv.org/abs/2001.05450
%X 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.
@misc{toen2020algebraic,
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 $\geq
2$, 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.},
added-at = {2020-05-22T10:27:47.000+0200},
author = {Toën, Bertrand and Vezzosi, Gabriele},
biburl = {https://www.bibsonomy.org/bibtex/2d2ad3560ce508e03ac937cb3b71f3809/simonechiarello},
description = {Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence},
interhash = {e815baa4bf612c9f4b907396c8871d12},
intrahash = {d2ad3560ce508e03ac937cb3b71f3809},
keywords = {Riemann-Hilbert},
note = {cite arxiv:2001.05450Comment: Added a few results. Submitted version},
timestamp = {2020-05-22T10:27:47.000+0200},
title = {Algebraic foliations and derived geometry: the Riemann-Hilbert
correspondence},
url = {http://arxiv.org/abs/2001.05450},
year = 2020
}