Narrow Results

We get boundedness for certain families of foliations. This is attained after proving a kind of foliated Arakelov inequality.

We extend the classification of complete polynomial vector fields in two complex variables given by Brunella to cover the case of holomorphic (non-polynomial) vector fields whose underlying foliation is still polynomial.

We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.

Let X be a projective manifold containing a quasi-line l. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper, we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety.

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ². As a sample of our results, we prove that N is biholomorphic to ℂ² if H²(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form where λ₁/λ₂ ∈ ℚ₊. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

Let $X$ be a compact Kähler manifold and let $T$ be a foliation cycle directed by a transversely Lipschitz lamination on $X$. We prove that the self-intersection of the cohomology class of $T$ vanishes as long as $T$ does not contain currents of integration along compact manifolds.

As a consequence, we prove that transversely Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliation cycles except those given by integration along compact leaves.

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family ${\{{\mathrm{\Omega }}^t\}}_{t\in D}$ of integrable one-forms ${\mathrm{\Omega }}^t$ defined in a neighborhood $U\subset {ℂ}^n$ of the initial singular point, and parametrized by the disc $D\subset ℂ$. The initial foliation is defined by ${\mathrm{\Omega }}^0$. The second type, more restrictive, is given by an integrable holomorphic one-form $\mathrm{\Omega }(x,t)$ defined in the product $U\times D\subset {ℂ}^n\times ℂ$. Then, the initial foliation is defined by the slice restriction $\mathrm{\Omega }(x,0)$. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by $df=0$ for some germ of holomorphic function $f\in {{𝒪}}_n$ at the origin $0\in {ℂ}^n,n\ge 3$. We assume that the germ $f$ is irreducible and that the typical fiber of $f$ is simply-connected. This is the case if outside of a dimension $\le n-3$ analytic subset $Y\subset {ℂ}^n$, the analytic hypersurface $X_f:(f=0)$ has only normal crossings singularities. We then prove that, if cod sing $\mathrm{\Omega }(x,0)\ge 2$ then the (germ of the) developing foliation given by $\mathrm{\Omega }(x,t)=0$ also exhibits a holomorphic first integral. For the general case, i.e. cod sing $\mathrm{\Omega }(x,0)\ge 1$, we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations ${\{{{ℱ}}_t\}}_{t\in ℂ,0}$, of a local pencil ${{ℱ}}_0:\frac{f}{g}=\mathrm{\text{constant}}$, for $f,g\in {{𝒪}}_n$. For dimension $n=2,$ we consider $f=x,g=y$. For dimension $n\ge 3,$ we assume some generic geometric conditions on $f$ and $g$. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type $\widehat{F}=\frac{f^{1+\widehat{\lambda }(t)}}{g^{1+\widehat{\mu }(t)}}{e}^{Ĥ(x,y,t)}$ with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form $M=\frac{f}{g}{e}^{P(t)+H(x,y,t)}$ with some additional properties, provided that for $n=2$ the axes remain invariant for the foliations ${{ℱ}}_t$.

In this work, we extend the residue theory from flag of holomorphic foliations to flag of holomorphic distributions and we provide an effective way to calculate this class in certain cases. As a consequence, we show that if we consider a flag ${ℱ}=({{ℱ}}_1,{{ℱ}}_2)$ of holomorphic distributions on ${\mathbb{P}}^3$, we get a relation between the degrees of the distributions in the flag, the tangency order of distributions, the Euler character characteristic and the degree of the curve $C$.

This paper is devoted_to the study of a system of differential equations: