Felipe Cano, Jean François Mattei, Marianna Ravara-Vago
It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}. Here, we state and prove a generalization of this property to any ambient dimension.
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We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays.
We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.
We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the celebrated theorem of Zariski for analytic curves, we prove the equivalence of these notions. We give some applications in the study of topological equivalences of holomorphic foliations. In particular, we show that the nodal singularities and its eigenvalues in the resolution of a generalized curve are topological invariants.
In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the number of singularities, with multiplicities, contained in the invariant quasi-smooth hypersurfaces; Poincare problem on weighted projective plane and the number of the hypersurfaces, of a degree fixed, invariant by a foliation on $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ which does not admit a rational first integral.
Holomorphic foliations tangent to Rolle-pfaffian hypersurfaces
本文研究复平面原点处满足Rolle-Khovanskii条件的与Pfaffian超曲面相切的全纯叶状结构芽，证明其由闭亚纯1-形式定义，并完成奇点消解中简单模型的分类。
Nilpotent Holomorphic Foliations on $(\mathbb{C}^n,\mathbf{0})$
本文研究复数维$(\mathbb{C}^n,\mathbf{0})$上的nilpotent光滑叶结构，通过分析其与Takens正常型的关系，提出新的准则以判断叶结构是否为广义超曲面类型。
We give a bounding of degree of quasi-smooth hypersurfaces which are invariant by a one dimensional holomorphic foliation of a given degree on a weighted projective space.
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse inva…
We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.
A min-max gap characterization of minimal foliations on the torus
本文扩展Mather引入的能量到Almgren-Pitts极小理论中，研究环面的最小流形分解，并证明在通用度量下，存在间隙时存在非极小的最小流形。
We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.
In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M \subset \mathbb{P}^{n}$, with singular set of real dimension at most $2n-4$ and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in $\mathbb{P}^{n}$. As a consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on $\mathbb{P}^{n}$ - a singular real analytic foliation whose leaves are i…
Let $\cal{F}$ be a regular codimension 1 holomorphic foliation on a compact K\" ahler manifold. One assumes in addition that $\cal{F}$ possesses a transverse invariant positive current. The aim of this paper is to establish the following alternative: - There exists an invariant hypersurface. - The foliation admits a transverse invariant hermitian metric with constant curvature
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…
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