Research Papers

Starting from 2009, I decided to keep track of the years in which the papers were originally written. Perhaps there are  some imprecisions before 2005. 

56. Codimension one foliations in positive characteristic. We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.
joint with W. Mendson
To appear in the Journal of the Institute of Mathematics of Jussieu

---

55. Closed meromorphic 1-forms. We review properties of closed meromorphic 1-forms and of the foliations defined  by them. We present and explain classical results from foliation theory, like index theorems, existence of separatrices, and resolution of singularities under the lenses of the theory of closed meromorphic 1-forms and flat meromorphic connections. We apply the theory to investigate the algebraicity separatrices in a semi-global setting (neighborhood of a compact curve contained in the singular set of the foliation), and the geometry of smooth hypersurfaces with numerically trivial normal bundle on compact Kähler manifolds.
Submitted

54.  Numerically nonspecial varieties. Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves i.e. rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Campana raised the question if one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the (split) function field setting. Finally, we conclude with a few comments for higher codimensional foliations which may provide some evidence towards a generalization of the aforementioned results.
Compositio Mathematica 158 (2022), 1428–1447.

---

53. A global Weinstein splitting theorem for holomorphic Poisson manifolds. We prove that if a compact Kähler Poisson manifold has a symplectic leaf with finite fundamental group, then after passing to a finite étale cover, it decomposes as the product of the universal cover of the leaf  and some other Poisson manifold.  As a step in the proof, we establish a special case of Beauville's conjecture on the structure of compact Kähler manifolds with split tangent bundle.

52. Codimension one foliations of degree three on projective spaces. We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension at least 3, extending a result by Loray, Pereira, and Touzet for degree three foliations on projective 3-space. We show that the space of codimension one foliations of degree three  has exactly 18 distinct irreducible components parameterizing foliations without rational first integrals, and at least 6 distinct irreducible components parameterizing foliations with rational first integrals.

48. Failure of the Brauer-Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell. Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.

45. Deformation of rational curves along foliations. Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms.We prove that codimension one foliations having a tangential foliation with at least one non-algebraic leaf are transversely homogeneous with structure group determined by the codimension of the non-algebraic leaf in its Zariski closure. As an application,we provide a structure theorem for degree three foliations on the projective space of dimension three.

43. Effective algebraic integration in bounded genus. We introduce and study birational invariants for foliations on projective surfaces built from the adjoint linear series of positive powers of the canonical bundle of the foliation. We apply the results in order to investigate the effective algebraic integration of foliations on the projective plane. In particular, we describe the Zariski closure of the set of foliations on the projective plane of degree d admitting rational first integrals with fibers having geometric genus bounded by g.

42. Toward effective liouvillian integration. We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of  degree comparatively small with respect to the degree of the foliation. We present a similar result for foliations with intermediate Kodaira dimension admitting a rational first integral.

41. Compact leaves of codimension one holomorphic foliations on projective manifolds. This article studies codimension one foliations on projective manifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holonomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.

40. Extactic divisors for webs and lines on projective surfaces. Given a web (multi-foliation) and a linear system on a projective surface we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the number of lines and of disjoint lines contained in a projective surface and tangent to a contact distribution.

39. Smooth foliations on homogeneous compact Kahler manifolds. We study smooth foliations of arbitrary codimension on homogeneous compact K\"ahler manifolds. We prove that smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations and classify the smooth foliations with all leaves analytically dense on compact homogeneous Kahler manifolds. Both results are builded upon a (rough) structure Theorem for smooth foliations on compact homogeneous Kahler manifolds obtained by comparison of the foliation and the Borel-Remmert decomposition of the ambient.

38. Webs invariant by rational maps on surfaces. We prove that under mild hypothesis rational maps on a surface preserving webs are of Latt\`es type. We classify endomorphisms of P^2 preserving webs, extending former results of Dabija-Jonsson.

37. Representations of quasiprojective groups, flat connections and transversely projective foliations. The main purpose of this paper is to provide a structure theorem for codimension one singular transversely projective foliations on projective manifolds. To reach our goal, we firstly extend  Corlette-Simpson's classification of rank two representations of fundamental groups of quasiprojective manifolds by dropping the hypothesis of quasi-unipotency at infinity. Secondly  we establish an analogue classification for rank $2$ flat meromorphic connections. In particular, we prove that a rank $2$ flat meromorphic connection with irregular singularities having non trivial Stokes projectively factors through a connection over a curve.

36. A characterization of diagonal Poisson structures. The degeneracy locus  of a generically symplectic Poisson structure on a Fano manifold is always a singular hypersurface. We prove that there exists just one family of generically symplectic Poisson structures in Fano manifold with cyclic Picard group having a reduced simple normal crossing degeneracy locus.
joint with Renan Lima
Bulletin of London Mathematical Society (2014) 46 (6), 1203-1217.

---

35. Polynomial completion of symplectic jets and surfaces containing involutive lines. Motivated by work of Dragt and Abell on accelerator physics, we study the completion of symplectic jets by polynomial maps of low degrees.  We use Andersén-Lempert Theory to prove that symplectic completions always exist, and we prove the degree bound conjectured by Dragt and Abell in the physically relevant cases.  However, we disprove the degree bound for $3$-jets in dimension 4.  This follows from the fact that if E is the disjoint union of r=7 involutive lines in projective space, then E is contained in a degree d=4  hypersurface (Todd).  We give two new proofs of this fact, and finally we show that if (r,d) differs from (7,4)  then the conjecture holds true.
joint with Erik Løw, Han Peters, Erlend F. Wold
Mathematische Annalen 364 (2016), no. 1-2, 519-538.

---

34. Transversely affine foliations on projective manifolds.We describe the structure of singular transversely affine foliations of codimension one on projective manifolds with zero first Betti number. Our result can be rephrased as a theorem on rank two reducible flat meromorphic connections.

33. Foliations with vanishing Chern classes. In this paper  we aim at the description of  foliations  having tangent sheaf  having  c_1 = c_2 = 0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure  of a general  leaf of  the foliation is an  Abelian variety. It turns out that the analytic type of the Zariski closures of leaves may vary from leaf to leaf.  We discuss how this variation is related to arithmetic properties of the tangent sheaf of the foliation.

32. Foliations with trivial canonical bundle on Fano 3-folds. We classify the irreducible components of the space of foliations on Fano $3$-folds with rank one Picard group. As a corollary we obtain a classification of holomorphic Poisson structures on the same class of $3$-folds.
joint with F. Loray, F. Touzet
Mathematische Nachrichten Volume 286 Issue 8-9, pp 921-940.

---

31. Singular foliations with trivial canonical class.  This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on projective manifolds.
joint with F. Loray, F. Touzet
Inventiones Mathematicae 213 (2018), no. 3, 1327-1380. (Springer link)

30. Rigid flat webs on the projective plane.  This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Legendre transform, is an avatar of classical projective duality in the realm of differential equations. We show that the Legendre transform of what we call reduced convex foliations are webs with zero curvature, and we exhibit  a countable infinity  family of convex foliations which give rise to a family of webs with zero curvature not admitting non-trivial deformations with zero curvature.
joint with D. Marin
Asian Journal of  Mathematics 17 (2013), no. 1, 163-192.

---

29. Resonance webs of hyperplane arrangements. Each irreducible component of the first resonance variety of a hyperplane arrangement  naturally determines a codimension one foliation on the ambient space. The superposition of  these foliations define what we call the resonance  web of the arrangement. In this paper we initiate the study of these objects  with emphasis on their spaces of abelian relations.

2009

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

27. Germs of integrable forms and varieties of minimal degree. We study the subvariety of  integrable 1-forms in a finite dimensional vector space W of germs of 1-forms. We prove that the irreducible components with dimension comparable with the rank of W are of minimal degree. 

26. Foliations invariant by rational maps.  We give a classification of pairs (foliation, rational map)  where the foliation is a  singular holomorphic foliation on a projective surface and the rational map is a non-invertible dominant rational map preserving the foliation. 

2008

2007

2005

2004

2003

2002

2001

2000