The characteristic variety of a generic foliation

Abstract

Abstract. We confirm a conjecture of Bernstein-Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.

On the density of strongly minimal algebraic vector fields

“…To study the structure of the invariant horizontal subvarieties of the first projective prolongation of an algebraic vector field, our primary ingredient is the description of the characteristic foliation of a generic foliation by curves obtained by Pereira in [Per12].…”

“…The horizontal subvarieties of type (I) of the first projective prolongation are studied in subsections 4.1 and 4.2. Under the assumptions (a) and (b), we are able to apply the results of [Per12] to conclude that the canonical invariant hypersurface is the only horizontal subvariety of type (I). The horizontal subvarieties of type (II) are studied in subsections 4.3 and 4.4 using an argument -based on the study of the tangency locus of an invariant distribution of codimension one with the foliation F (v)similar to the one used in dimension two in [Jao21].…”

“…Acknowledgments. The author would like to thank J. V. Pereira for several discussions during the summer school Feuilletages et Géométrie Algébrique in Grenoble in 2019 as well as during the workshop Foliations in Algebraic and Birational Geometry in Lausanne in 2022 on his paper [Per12] and several additional arguments used in this article. The author is also very grateful to Jean-Benoît Bost who suggested to him during his PhD to study this problem using tools from foliation theory.…”

“…The subsections 2.2 to 2.4 introduce some vocabulary on twisted algebraic vector fields that will be used in Section 5. All the definitions and results contained in these subsections are taken from [CP06], [Cou11] and [Per12]. Finally, in subsections 2.5 and 2.6, we recall some classical terminology on distributions and foliations in the setting of differential algebra from [Jao20b] that will be used in Sections 3 and 4.…”

“…where θ is any (local) lift of θ to Θ X defines a partial connection on the normal bundle N F of F along the foliation F called Bott's partial connection (see [Per12], Section 4.2).…”