On the moduli space of certain smooth codimension one foliations of the 5-sphere by complex surfaces
Laurent Meersseman, Alberto Verjovsky
研究了5-球面光滑叶理的复流形变形空间
We study the space of deformations of a smooth foliation of the 5-sphere by complex manifolds
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Codimension two polar homogeneous foliations on symmetric spaces of noncompact type
本研究针对不可约非紧型对称空间上余维数为2的齐性极叶状结构，按轨道等价性完成分类，明确其要么为超极叶状结构，要么是秩一对称空间上同类叶状结构的典范扩张。
Jacobi fields and the stability of minimal foliations of arbitrary codimension
本文研究了任意余维极小叶理的叶的稳定性。我们还研究了Jacobi场与保持叶理的向量场之间的关系，并将这些结果应用于Killing场。
We extend the classification of homogeneous codimension-one foliations on irreducible Riemannian symmetric spaces of noncompact type obtained by Berndt and Tamaru to the reducible case, thus completing it for all noncompact symmetric spaces.
Deformations of pairs of codimension one foliations
本文研究余维一叶状结构对线性形变为切触对的问题，并给出主要结果的应用。
An example concerning the Theory of Levels for codimension-one foliations
我们给出一个余维1叶理的例子，它是类C^1的横截，并且不满足“局部极小集”性质。
Euclidean hypersurfaces with a totally geodesic foliation of codimension one
对欧几里得空间中具有余维为1的完全叶的全测地叶理的超曲面进行了分类。特别地，我们证明了具有余维1的完整轮廓的旋转超曲面的特征在于它们的翘曲乘积结构。还考虑了问题的本地版本。
We discuss the decomposition of degree two basic cohomology for codimension four taut Riemannian foliation according to the holonomy invariant transversal almost complex structure J, and show that J is C pure and full. In addition, we obtain an estimate of the dimension of basic J-anti-invariant subgroup. These are the foliated version for the corresponding results of T. Draghici et al.
A preparation theorem for codimension one foliations
在胶合叶状复流形之后，我们得到了余维为1的叶状流形和平面向量场（在实或复的情况下）的奇点的一个类分解定理。在没有计算的情况下，我们检索和改进的结果Levinson-Moser的功能，Dufour Zhitomirskii的非退化余维1叶理（证明反过来的解析性），Strozyna-Zoladek的非退化平面向量场和Brjuno-Ecalle鞍结叶理在平面上。
We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.
Polar foliations on quaternionic projective spaces
本文对四元数射影空间$\mathbb H P^n$上的余维$q$的不可约极叶理进行了分类，对所有的$(n,q)\neq(7,1)$ .我们证明了任何余维的所有不可约极叶理（分别为）。余维1）是齐次的，当且仅当$\mathbb H P^n$是素数（分别为）。$n$是偶数或$n=1$）。这表明存在余维为1或更高的非齐次例子。
Transversely affine holomorphic foliations of arbitrary codimension - I
研究具有仿射齐次横截结构的全纯叶理。我们给出了一个友好的刻画的情况下，横向仿射叶理的矩阵值对的微分形式。这自然导致研究的情况下，叶理与奇点。第一扩张定理，然后证明在一般的奇异性框架。
The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such resu…
Harmonic Morphisms on Lie groups and Minimal Conformal Foliations of Codimension two
本文研究配备左不变半黎曼度量的李群上由半单子群生成的余维二左不变共形叶状结构，证明其极小性，并在黎曼情形下给出全测地条件。
We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds.
Let $\mathscr{F}$ be a singular holomorphic foliation, of codimension $k$, on a complex compact manifold such that its singular set has codimension $\geq k+1$. In this work we determinate Baum-Bott residues for $\mathscr{F}$ with respect to homogeneous symmetric polynomials of degree $k+1$. We drop the Baum-Bott's generic hypothesis and we show that the residues can be expressed in terms of the Grothendieck residue of an one-dimensional foliation on a $(k+1)$-dimensional disc transversal to a $(k+1)$-codimensional component of the singular set of $\mathscr{F}$. Also, we show that Cenkl's algor…
Manifolds that are not leaves of codimension one foliations
我们提出了一个新的开流形，它不同胚于紧致流形的任何C^0余维的叶。其中有5维或更高维的单连通流形，它们在同伦或同调中是非周期的，即在它们的2维同伦或同调群中。
A class of codimension one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally non-degenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson type submanifold. In particular we deduce that their leaf spaces are homeomorphic to leaf spaces of 3-dimensional taut foliations. We also introduce surgery constructions to show that this class of foliations is broad enough. Our techniques come mainly from symplectic geometry.
Riemannian manifolds not quasi-isometric to leaves in codimension one foliations
每一个维数大于1的开流形L都有完备的黎曼度量g，g具有有界几何，使得（L，g）不拟等距于闭流形的余维数为1的叶的叶子.因此，没有条件的局部几何（L，g）足以使它准等距的一个叶，这样的叶理。我们引入了“有界同调属性”，（L，g）的半局部属性，这是必要的，它是一个叶在一个紧凑的流形余维1，直到准等距。一个重要的步骤涉及部分推广诺维科夫闭叶定理更高的层面。
A-Foliations of codimension two on compact simply-connected manifolds
该研究证明二维余维数的紧致单连通微分流形上的奇异Riemann叶结构可通过光滑有效$n$- торus作用给出，解决了关于exotic торi叶结构存在的问题。
The Noncommutative Foliation Invariant (NCFI): extension to the odd codimension case and computed examples
本文扩展奇数余维叶状结构的非交换叶状不变量定义，计算了多种叶状结构示例，包括纤维化、克罗内克等，还证明了一般结果，在偶数和奇数余维情形下分别阐述了不变量计算方式及与相关类的关系。
For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic cohomological group is of zero dimension, we can give an estimate for the index of transversal Dirac operator of a foliated spin structure. Furthermore, under the condition that $H^\pm_b(M)=1$, we show the vanish…
Isoparametric foliations of $S^n\times S^n$ of codimension $1$
本文给出 \(S^n\times S^n\) 等参叶状结构新例族，建立其基本性质，还对双次数为 \((1,1)\) 和 \((2,2)\) 的双齐次等参多项式及特定 \(n\) 的 \(S^n\times S^n\) 上的等参叶状结构进行分类。
In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\bar{HM}(M,F,\mfs;Γ),~\hat{HM}(M,F,\mfs;Γ), ~\widecheck{HM}(M,F,\mfs;Γ)$, for each transverse \spinc structure $\mfs$, where $Γ$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologie…
Codimension one foliations with Bott-Morse singularities I
研究了具有非空紧奇异集$\sing(\fa)$的定向闭流形$M$上的余维1（横定向）叶理$\fa$，其中$\fa$是由Bott-Morse函数局部定义的.证明了：如果$\fa$在每个奇点的横截型是中心，且$\fa$有一个基本群有限的紧叶，或$\sing(\fa)$的一个分支有余维$\ge 3$和有限的基本群，则$\fa$的所有叶都是紧的，是同构的，$\sing(\fa)$由两个连通分支组成，并且存在Bott-Morse函数$f:M \to [0,1]$，使得$f\colon M \setminus \sing(\fa) \to (0,1)$是定义$\fa$和$\sing(\fa) = f^{-1}(\{0,1\})$的纤维束。这产生了拓扑描述的类型的叶子出现在这些叶理，也是类型的流形承认这样的叶理。这些结果统一和推广了上齐性等距作用的著名结果和中心型Morse奇点叶理的Reeb定理。在这种情况下，$\fa$的每个叶是$\sing(\fa)$的每个分量上的球纤维丛。
Following Losik's approach to Gelfand's formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, the Mizutani-Morita-Tsuboi Theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. I…
A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold
设$(M^{n},g)$是一个闭的、连通的、定向的、$C^{\infty}$的、黎曼的、n-流形，它具有一个横向定向的叶理$\boldkey F$ .我们证明了，如果$\lbrace X,Y \rbrace$是基本向量场，则当$κ\wedge χ_{\boldkey F}$是封闭的（可能为零）de Rham上同调（p+1）-形式时，$[X,Y]$的叶分量$\Cal{V}[X,Y]$的叶发散为零.这里$κ$是叶理${\boldkey F}$的平均曲率的一种形式，而${χ_{\boldkey F}}$是它的特征形式。在余维2的情况下，当且仅当$κ$水平闭合时，$κ\wedge χ_{\boldkey F}$闭合。在一定的限制条件下，给出了$κ\wedge{χ_{\boldkey F}}$是调和的充要条件.作为应用，我们给出了某些闭三维流形是局部黎曼积的一个刻划。我们证明了常曲率流形上的具有不可积横截分布的、叶维数大于或等于2的全脐叶的类Einstein-like叶理是全测地线的。
Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry
本研究定义了紧复环面上推广Ghys余维一情形的高余维非奇异全纯湍流叶状结构，证明其所有横截全纯Cartan几何均平坦，并给出了该几何的唯一性结果。
Minimal foliations, codimension-one stable norms, and a question of Bangert
该研究计算环面上等距群余维一度量的余维一稳定范数，对Bangert的问题给出否定答案，在三维环面构造无限维非平坦度量族，证明欧氏稳定范数与体积数据在立方平坦度量附近非局部单射。
Stability of Diophantine Minimal Foliations in Arbitrary Codimension
该研究将Moser的极小超曲面叶状结构稳定性定理推广到任意余维，证明标准环面上满足丢番图条件的平面诱导的仿射叶状结构，在欧氏度量的小周期扰动下仍以极小子流形构成的光滑共轭叶状结构存在，采用Hamilton的Nash-Moser隐函数定理完成证明。
Foliations transverse to a closed conformal vector field
本文研究黎曼流形上横截于闭共形向量场的余维一叶状结构，提出Montiel叶状结构概念，刻画了全测地叶片条件及极小与常平均曲率叶状结构的几何性质。
Complex codimension one singular foliations and Godbillon-Vey sequences
设F是紧致复流形M上的余维奇异全纯叶理。设M上存在一般横截于F的亚纯向量场X.然后，我们证明了F是代数流形N上代数叶理的亚纯拉回，或者F是紧致超曲面外的横射，改进了我们以前的工作（见第1版）。这样一个向量场保证了叶理F的整体亚纯Godbillon-Vey序列的存在性。我们得到了这个序列保证这种选择的充分条件。例如，如果存在一个有限的Godbillon-Vey序列，或者如果Godbillon-Vey不变量为零，则F是曲面上叶理的拉回，或者F是横向投射的。我们用许多例子来说明这些结果。
In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for th…
Isoparametric foliations on complex projective spaces
对复射影空间CP^n上任意余维q的不可约等参叶理进行了分类，但n=15和q=1除外.值得注意的是，有一些非全等的例子，在霍普夫映射下，它们被拉回到球面上的全等叶理。此外，还存在许多非均匀的等参叶理，甚至有更高的余维。事实上，复射影n-空间上的每个不可约等参叶理都是齐次的当且仅当n+1是素数。本文的主要工具是研究复射影空间上具有闭叶的奇异黎曼叶理。这种方法是基于一种图，它推广了内对称空间的扩展Vogan图。
In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an $n$-dimensional compact, complete, and oriented affine manifold endowed with a codimension $1$ linear foliation ${\cal F}$ is homeomophic to the $n$-dimensional torus if the leaves of ${\cal F}$ are simply connected. Let $(M,\nabla_M)$ be a $3$-dimensional compact affine manifold endowed with a codimension $1$ linear foliation. We prove that $(M,\nabla_M)$ has a finite cover which is homeomorphic to the total space of a bund…
The diffeomorphism group of a Lie foliation
我们明确计算的几种类型的线性叶理（密集的叶子）的环面$T^n$，$n\geq 2$，即余维一叶理，流，和所谓的非二次叶理的同胚群。我们特别表明，非二次叶理是刚性的，在这个意义上说，他们不承认横向同胚以外的$\pm \id$和翻译。计算是一个通用的公式，我们证明了一个紧凑的流形上的任何李叶理稠密的叶的同胚群的应用。我们的结果推广了P.Donato和P.Iglesias对$T^2$，P.Iglesias和G. Lachaud对$T^n$，$n\geq 2$和B上的余维1叶理进行了研究. Herrera为卓越的叶理。本文的理论背景是J.M. Souriau的空间理论
Polar foliations on symmetric spaces
证明了在不可约紧致对称空间中余维至少为3的极叶理是超极的，除非对称空间的秩为1。对于紧致型的可约对称空间，我们得到了极叶理的分解结果。
Classification of holomorphic foliations on Hopf manifolds
本文对一类Hopf流形上的一维和余维非奇异全纯叶理进行了分类。更一般地，我们证明了所有的非奇异余维1分布的中介或一般的Hopf流形是可积的，有全纯积分第一。同时，我们证明了关于Hopf流形上奇异全纯分布的一些结果.
Codimension one symplectic foliations and regular Poisson structures
在这个简短的说明中，我们给出了一个完整的特征的某类紧corank一Poisson流形，那些配备了一个封闭的形式定义的辛叶理和一个封闭的两个形式扩展的辛形式在每个叶子。如果这样的流形有一个紧叶，则所有的叶都是紧的，并且该流形是紧叶的映射环面。这些流形和它们的正则泊松结构允许一个扩展作为泊松b-流形的临界超曲面，我们将在后面的论文中考虑。
Extrinsic geometric flows on foliated manifolds, III
本文研究了具有含时黎曼度量的余维一叶理的几何性质。工作开始与公式有关变形的几何量作为黎曼度量变化沿叶的叶理。然后介绍了基于叶理的第二基本型的外生几何流。在适当的假设下，这种演变产生的二阶抛物型偏微分方程，其中的存在/uniquenes和在某些情况下收敛的解决方案。应用于规定余维一叶理的中曲率函数的问题，以及调和叶理和脐叶理的例子（例如，叶状表面）和扭曲的产品度量。
This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topological classes in terms of the cohomology of a new algebraic object that we call group-graph. This moduli space may be an infinite dimensional functional space but under generic conditions we prove that it has finite dimension and we describe its algebraic and topological struct…
Differentiable classification of 4-manifolds with singular Riemannian foliations
本文首先证明了：任意闭单连通4-流形，若其可分解为两个秩大于1的圆盘丛，则它与标准椭圆4-流形$\mathbb{S}^4$，$\mathbb{CP}^2$，$\mathbb{S}^2\times\mathbb{S}^2$，或$\mathbb{CP}^2\#\pm \mathbb{CP}^2$之一同构.作为应用，我们证明了任何允许非平凡奇异黎曼叶状的闭单连通4-流形都同构于标准$\mathbb{S}^4$，$\pm\mathbb{CP}^2$和$\mathbb{S}^2\times\mathbb{S}^2$的连通和.作为一个副产品，得到了所有闭单连通4-流形上余维为1的奇异黎曼叶理的分类。特别地，恰好有3个余维为1的非齐次奇异黎曼叶理，补充了一个4-流形的上齐次列表。
Characteristic classes of foliations via SAYD-twisted cocycles
我们之前已经证明，任何李代数的截断Weil代数都是具有非平凡系数的霍普夫循环型复形。本文应用这一结果将横定向叶理的特征类转化为广群作用代数的循环上同调。我们在余维1的结果与Connes-Moscovici唯一的显式计算相匹配。在余维为2的情形下，我们构造性地给出了一个显式计算，并将横截基本类、Godbillon-Vey类和其它四个剩余类表示为群胚作用代数上的循环余圈.在这个新的特征映射中，主要负责的对象是与余维度相同的SAYD-扭曲循环上圈。我们通过引入等变的Hopf-循环上同调和等变的上积构造了这样一个上圈.
A conformal integral invariant on Riemannian foliations
设$M$是一个闭流形，它有一个余维为$q\geq 2$的叶状结构$\mathcal{F}$和一个类三角形度量$g_0$ .令$[g_0]_B$是类似于矩阵的度量的空间，其仅在水平方向上与$g_0$相差正基函数的倍数。假设$Y$是横截共形向量场，并且$(M,\mathcal{F},g_0)$的叶的平均曲率为零。我们表明，积分$\int_MY(R^T_{g^T})dμ_g$是独立的选择$g\in [g_0]_B$，其中$g^T$是横向度规引起的$g$和$R^T$是横向标量曲率。此外，如果$q\geq 3$，我们有任何$g\in [g_0]_B$的$\int_MY(R^T_{g^T})dμ_g=0$。但存在余维2极小黎曼叶理$(M,\mathcal{F},g)$和横截共形向量场$Y$使得$\int_MY(R^T_{g^T})dμ_g\neq 0$ .因此，它是一个非平凡的障碍横Yamabe问题上的最小黎曼叶理的余维2。
We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,α+β)$. Using a unit vector field $ν$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(α,β)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $β$ which simplify derivations, we express them in terms of corresponding invariants arising …
Let F be a codimension-one, C^2-foliation on a manifold M without boundary. In this work we show that if the Godbillon--Vey class GV(F) \in H^3(M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C^1-dynamical systems, and does not use the classification theory of C^2-foliations. We first prove that for a codimension--one C^1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resil…
Feuilletages totalement geodesiques, flots riemanniens et varietes de Seifert
研究了Lorentz流形上的全测地余维1光滑叶理。我们特别感兴趣的黎曼流和测地叶理之间的关系。我们证明，直到2-覆盖，任何塞弗特丛承认这样的叶理。
Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that to…
In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurf…
Lorentzian foliations on 3-manifolds
研究了闭三维流形上的横洛伦兹叶理。我们在完备性假设下对它们进行了分类，并推导出余维1测地完备类时全测地叶理的对偶分类。此外，我们提供了一个例子的洛伦兹叶状的紧3-流形是等度连续的流形的一个适当的稠密开子集，因此不是横向完整的。
We apply gauge theory to study the space $F_k(M)$ of smooth codimension-$k$ framed foliations on a smooth manifold $M$. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains $F_k(M)$ as a subspace. The notion of holonomy is naturally extended to this moduli space and the cohomology theory associated with points of this moduli space which correspond to non-singular foliations coincides with Bott cohomology. The quotient of the moduli space under concordance is identified as the space of homotopy classes of…
The aim of the paper is to prove the following result concerning moduli of curve families in the Heisenberg group. Let $Ω$ be a domain in the Heisenberg group foliated by a family $Γ$ of legendrian curves. Assume that there is a quadratic differential $q$ on $Ω$ in the kernel of an operator defined in \cite{Tim2} and every curve in $Γ$ is a horizontal trajectory for $q$. Let $l_Γ: Ω\rightarrow ]0,+\infty[$ be the function that associates to a point $p\in Ω$, the $q$-length of the leaf containing $p$. Then, the modulus of $Γ$ is \[ M_4 (Γ) = \int_Ω\frac{|q|^2}{(l_Γ) ^4} \mathrm{d} L^3.\]
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 …
Foliation surgeries and the concordance groups of foliated spheres
本文通过叶状手术和连通和构造了叶状球的同调群，并给出了显式海弗利格结构及其Godbillon-Vey数的映射结果。
Normal forms of foliations and curves defined by a function with a generic tangent cone
我们首先描述的本地和全球的芽叶理定义的解析函数在两个变量的p横向光滑分支，并与积分的多重性（在单值全纯的情况下）或复杂的多重性（在多值“达布”的情况下）的模空间。我们在每个类中指定标准形式。然后在这些模空间上研究了由以下等价关系导出的分布C：两点等价当且仅当对应的叶理在解析共轭下具有相同的解析不变曲线。因此，C的叶空间是曲线的模空间。我们证明了C是有理可积的。这些有理积分给出了这些一般平面曲线的完整的不变量系统，它扩展了分支之间的公知交比。
Masur's divergence states that the horizontal foliation of translation surfaces is uniquely ergodic if the geodesic flow is recurrent on the moduli space. This established a relationship between geometrical properties of foliations and the dynamics on the moduli space. In this paper, we extend this theorem to complex torus and Kummer surfaces. We define and calculate horizontal foliations and the corresponding geodesic flow in the moduli space of Kähler metrics and prove that the horizontal foliation is uniquely ergodic if the geodesic flow is recurrent. We also find the a necessary algebraic …
Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains
本文建立了Kähler流形上整体生成的全纯向量丛的截面，关于非退化二次型的迷向截面和欧氏开域上的全测地叶理截面之间的关系.我们找到了全测地叶理起源于全纯向量丛的一个几何条件。对于余维为2的叶理，这种描述恢复了P.Baird和J.C.木材.起关键作用的泛对象是正交格拉斯曼。
Moduli Space of Dihedral Spherical Surfaces and Measured Foliations
本文研究二面角球面曲面，通过定义一对横向测度叶状结构来完全刻画该类曲面，并引入几何分解与形变方法，最终确定给定锥角和拓扑类型下模空间的维数。
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 $X$.
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