We show that the set of singular holomorphic foliations of the projective spaces with split tangent sheaf and with good singular set is open in the space of holomorphic foliations. As applications we present a generalization of a result by Camacho-Lins Neto about linear pull-back foliations, we give a criterium for the rigidity of $\mathcal L$-foliations of codimension $k \ge 2$ and prove a conjecture by Cerveau-Deserti about the rigidity of a codimension one $\mathcal L$-foliation of $\mathbb P^4$. These results allow us to exhibit some previously unknown irreducible components of the spaces of singular holomorphic foliations.
We study Segre varieties associated to Levi-flat subsets in complex manifolds and apply them to establish local and global results on the integration of tangent holomorphic foliations.
The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem of Enoki. We apply this result and the techniques involved in its proof to get a version of this theorem for stable varieties (the higher dimensional analogue of Deligne-Mumford's stable curves) and to prove the polystability with respect to any polarization of the tangent sheaf of a singular Calabi-Yau variety.
This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the vector field - for instance, that the final models belong to a family of vector fields whose linear parts have eigenvalues satisfying a condition of non-resonance - in order to conclude that the foliation is of complex hyperbolic type, that is, without saddle-nodes in its reduction of singularities. In the second part, we prove that a vector field that is tan…
Let $\mathcal{F}$ be a codimension one holomorphic foliation in $\mathbb{P}^n$, $n\geq 2$, leaving invariant a real-analytic Levi-flat hypersurface $M$ with regular part $M^{*}$. Then every leaf of $\mathcal{F}$ outside $\overline{M^{*}}$ accumulates in $\overline{M^{*}}$.
Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincaré mass of an extremal positive \ddc-closed current tangent to \Fc. If, moreover, there is no nonzero positive closed current tangent to \Fc, then we show that the Lyapunov exponent χ(\Fc) of \Fc, which is, by definition, the Lyapunov exponent of the unique normalized positive \ddc-closed cur…
Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.
We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension $2$ is birational to a Grass…
We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in $\mathbb{P}^2$.
Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work of Paun and Sibony, we study the intersection of Ahlfors current with the tangent bundle of the foliation, and derive some consequences. In particular, we introduce the definition of weakly reduced singularities for foliations by curves, which requires less work than the exact classification for foliations. Finally we discuss the strategy to prove the Green-…
We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leave…
This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally…
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one …
Beauville asked if a compact Kähler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.
In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. A…
This article is dedicated to the study of singular codimension $1$ foliations $\mathcal{F}$ on a simplicial complete toric variety $X$ and their pullbacks by dominant rational maps $φ:\mathbb{P}^n\dashrightarrow X$. First, we describe the singularities of $\mathcal{F}$ and $φ^*\mathcal{F}$ for a generic pair $(φ,\mathcal{F})$. Then we show that the first order deformations of $φ^*\mathcal{F}$ arising from first order unfoldings are the families of the form $φ_\varepsilon^*\mathcal{F}$, where $φ_\varepsilon$ is a perturbation of $φ$. We also prove that the deformations of the form $φ^*\mathcal{…
This paper stresses the strong link between the existence of partial holomorphic connections on the normal bundle of a foliation seen as a quotient of the ambient tangent bundle and the extendability of a foliation to an infinitesimal neighborhood of a submanifold. We find the obstructions to extendability and thanks to the theory developed we obtain some new Khanedani-Lehmann-Suwa type index theorems.
The leaves in singular holomorphic foliation theory are examples of quasi-analytic layers. In the first part of our publication we are concerned with a theory of these subjects. A quasi-analytic decomposition of a complex manifold is a decomposition into pairwise disjoint connected quasi-analytic layers. These are holomorphic foliations in the sense of P. Stefan and K. Spallek. A very different but more usual conception of holomorphic foliations is develloped by P. Baum and R. Bott. It is based on holomorphic sheaf theory. In the second part we study the relation between quasi-analytic decompo…
A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\…
Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $\mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient condi…
Let $M$ be a complex- or real-analytic manifold, $θ$ be a singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the class of singularities of $θ$, under the hypothesis that the considered class of singularities is invariant by $θ$-admissible blowings-up. In particular, if $θ$ is monomial, we prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the monomiality of the singular distribution $θ$.
In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize …
Let $X$ be a normal complex space such that the tangent sheaf $T_X$ is locally free and locally admits a basis consisting of pairwise commuting vector fields. Then $X$ is smooth.
We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally contained in a leaf. Let T be a harmonic current tangent to \Fc which does not give mass to any invariant analytic curve. Using the leafwise Poincaré metric, we show that H is integrable with respect to T. Consequently, we infer the existence of the Lyapunov exponent function of T.
Let $X$ be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of $X$: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover $X$ splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of $X$ according to its holonomy representation. In particular, we classify those $X$ which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Cal…
