S. Ivashkovich
The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell.
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Uniformizable foliated projective structures along singular foliations
研究了紧复流形上由曲线构成的叶形的可统一投影结构，发现奇点类型对结构存在性有严格限制，并证明了某些条件下叶形无法具有强可统一的投影结构。
Foliated affine and projective structures
本文形式化复流形上沿全纯叶状结构叶片的仿射与射影结构，给出奇点局部正规形式和紧致情形下的指标公式，并证明一般型正则叶状结构不承认叶状射影结构，最终分类了紧复曲面上的此类结构。
We consider closed positive currents invariant by a singular holomorphic foliation on an algebraic surface. We show that under some conditions the foliation must leave invariant an algebraic curve.
We study those real $\mathcal{C}^\infty$ foliations in complex surfaces whose leaves are holomorphic curves. The main motivation is to try and understand these foliations in neighborhoods of curves: can we expect the space of foliations in a fixed neighborhood to be infinite-dimensional, or are there some contexts under which every such foliation is holomorphic? We give some restrictions and study in more details the geometry of foliations whose leaves belong to a holomorphic family of holomorphic curves. In particular, we classify all real-analytic foliations on neighborhoods of curves which …
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
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.
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.
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.
Basic (Dolbeault) Cohomology of Foliated Manifolds with Boundary
本文为具基本边界流形上的Riemannian和Hermitian叶状结构发展了$L^2$理论，建立了基本上同调的分解、消灭、扭曲对偶与延拓定理，并证明了复类似物及研究了$\bar{\partial}_B$-方程的整体正则性。
We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.
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 …
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.
In this paper we study transversely holomorphic foliations of complex codimension one with some hypothesis on the transverse structure.
We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes.
Log Baum--Bott Residues for foliations by curves
本文研究了曲面的Log Baum--Bott余数，提出了新的指标与Poincaré问题的关系，并证明了Zariski--Lipman猜想的弱全局版本。
We classify nonsingular holomorphic foliations of dimension and codimension one on certain Hopf manifolds. More general, we prove that all nonsingular codimension one distributions on intermediary or generic Hopf manifolds are integrable and has holomorphic integral first. Also, we prove some results about singular holomorphic distributions on Hopf manifolds.
We show that the topological equivalence class of holomorphic foliation germs with an isolated singularity of Poincaré type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincaré type and discuss a conjecture on the classification in dimension $\geq 3$.
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 $\mathbb{P}^2$.
We investigate the accumulation to singular points of leaves of codimension one foliations whose normal bundle is ample, with emphasis on the nonexistence of Levi-flat hypersurfaces.
It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends to a $d$-web near a nondicritical singular point and admits a multiple-valued meromorphic first integral.
We classify nonsingular holomorphic distributions of arbitrary codimension on certain Hopf manifolds. We prove that all holomorphic distribution of codimension k on a generic Hopf manifold is induced by a mononial holomorphic k-form.
Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $Δ$ be a foliation on $X$ such that each leaf $ω\inΔ$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cuttin…
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
We prove that a transversely holomorphic foliation which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not of zero measure. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold, is finite provided that the set of periodic orbits is not of zero measure.
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.
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 $X$.
We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.
After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for non degenerate codimension 1 foliations (proving in turn the analyticity), Strozyna-Zoladek for non degenerate planar vector fields and Brjuno-Ecalle for saddle-node foliations in the plane.
In this paper, we study a notion of hyperbolicity for hyperbolicity foliations with 1-dimensional parabolic leaves, namely the non-existence of holomorphic cylinders along the foliation - holomorphic maps from $\D^{n-1} \times \C$ to the manifold sending each $\{*\} \times \C$ to a leaf. We construct a tensor, that is a leaf-wise holomorphic section of a bundle above the foliated manifold, which vanishes on an open saturated set if and only if there exists such a cylinder through each point of this open set. Thanks to this study, we prove that open sets saturated by compact leaves are union of…
We prove the following theorem for Holomorphic Foliations in compact complex kaehler manifolds: if there is a compact leaf with finite holonomy, then every leaf is compact with finite holonomy. As corollary we reobtain stability theorems for compact foliations in Kaehler manifolds of Edwards-Millett-Sullivan and Hollman.
In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this context .
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