%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 }