Marco Brunella
These are lecture notes for a course to be held. They provide a full discussion of certain analytic aspects of the uniformisation theory of (singular) holomorphic foliations by curves on compact Kaehler manifolds, with emphasis on their consequences on positivity properties of the corresponding canonical bundles.
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This paper deals with codimension one (may be singular) foliations on compact Kälher manifolds whose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a rigidly embedded invariant hypersurface.
We prove the existence of regular foliations with a prescribed tangency divisor in neighborhoods of negatively embedded holomorphic curves; this is related to a linearization theorem due to Grauert. We give also examples of neighborhoods which can not be linearized.
We present enumerative results for holomorphic foliations by curves on $\mathbb{P}^n$, $n\geq 3$, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa--Fernández-Pérez--Nonato Costa--Vidal Martins and Nonato Costa. We also present an enumerative result bounding the number of isolated singularities in a projective subvariety invariant by a holomorphic foliation by curves on $\mathbb{P}^n$ with a singularity of positive dimension.
Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space
该研究针对三维射影空间上无孤立奇点的曲线叶状结构，分类了3次及以下结构的余正规层与奇异概形的不变量，刻画了更高次的两类叶状结构的性质与模空间。
Neighborhoods of curves with a prescribed number of foliations
本研究针对n维射影空间中连通射影曲线的邻域问题，构造了恰好带指定数目余维1全纯叶状结构的n维非紧复流形，确定其亚纯函数域的可能形式，推广并改进了曲面情形的相关结果。
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-…
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $Σ$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $\mathcal F$ and $C_i$, assuming that $\mathcal F$ is special along the $C_i$. Allowing just one nonzero dimensional component on $Σ$, we also prove results on when the foliation happens to be determined by its singular locus.
On some indices of foliations and applications
本文研究了叶的某些指标及其应用，推导了Tjurina数的局部公式，并提供了关于叶乘数和不变曲线Milnor数的新的全局结果证明。
Example of a foliation for Ueda type one
本文构造了一个沿亏格 3 曲线的正则二维叶状结构实例，其 Ueda 类型为一且法丛阶为二，为相关分类问题提供了具体例证。
Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${mathbb A}^2_\k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field exte…
Formulae for indices of holomorphic foliations via reduction of singularities
本文通过奇点约简过程，推导出全纯叶系的多种指标公式，包括歧变向量、Milnor数、经典指数等，并统一了相关理论框架。
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 …
In this paper we study the formal classication of two-dimensional neighborhoods of genus g $\ge$ 2 curves with trivial normal bundle. We first construct formal foliations on such neighborhoods with holonomy vanishing along many loops, then give the formal/analytic classication of neighborhoods equipped with two foliations, and finally put this together to obtain a description of the space of neighborhoods up to formal equivalence.
Excess logarithmic residues for foliations by curves and applications
本文研究了一维全纯丛的过剩对数留数，通过比较对数正常sheaf与普通正常sheaf，测量局部变化，并推导出Chern数的全局留数公式及Poincaré型界，最后证明了提升后的叶形结构恢复了奇异性的log discrepancy。
Completely Integrable Foliations: Singular Locus, Invariant Curves and Topological Counterparts
本文研究余维数 q ≥ 2 的完全可积全纯叶状结构，证明其奇点附近要么正则、要么非孤立、要么存在无穷多分隔叶，并通过引入全和乐群等拓扑工具给出具有孤立分隔叶的完全可积奇点的拓扑刻画。
Let $C$ be a smooth elliptic curve embedded in a smooth complex surface $X$ such that $C$ is a leaf of a suitable holomorphic foliation of $X$. We investigate complex analytic properties of a neighborhood of $C$ under some assumptions on complex dynamical properties of the holonomy function. As an application, we give an example of $(C, X)$ in which the line bundle $[C]$ is formally flat along $C$ however it does not admit a $C^\infty$ Hermitian metric with semi-positive curvature.
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.
A higher-dimensional Van den Essen type formula for projective foliations and applications
该研究针对射影空间中沿光滑完全交奇异的一维全纯叶状结构，推导了高维Van den Essen型Milnor数变化公式，给出Milnor数下界，证明沿光滑曲线奇异叶状结构可经有限次曲线中心爆破化为重数1且一般点为初等奇点的叶状结构，并得到解消所需爆破次数的界。
A transfer principle: from periods to isoperiodic foliations
本研究分类了Hodge丛上等周期叶状结构的叶闭包并证明其遍历性，借助模空间Torelli覆叠上周期映射纤维的连通性推导结果，还得到椭圆曲线原始分支覆叠Hurwitz空间的拓扑推论，同时发展了增广Torelli空间理论。
We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
Given a projective symplectic manifold $M$ and a non-singular hypersurface $X \subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an étale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families o…
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…
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。
Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive ddc-closed (1,1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. This property is surprising because for most of suc…
Let $\mathcal{F}$ be a holomorphic foliation by curves defined in a neighborhood of $0$ in $\mathbb{C}^n$ ($n\geq 2$) having $0$ as a weakly hyperbolic singularity. Let $T$ be a positive harmonic current directed by $\mathcal{F}$ which does not give mass to any of the $n$ coordinate invariant hyperplanes $\{z_j=0\}$ for $1\leq j\leq n.$ Then we show that the Lelong number of $T$ at $0$ vanishes. Moreover, an application of this local result in the global context is given. We discuss also the relation between several basic notions such as directed positive harmonic currents, directed positive d…
For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.
We classify two dimensional neighborhoods of an elliptic curve C with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having C as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discussanalytic equivalence and show, for each formal model, that the corresponding moduli space is infinite dimensional.
The Poincaré problem for reducible curves
本文研究了复曲面中局部全纯叶状结构的庞加莱问题，基于不变曲线芽数据给出了叶状结构重数的精确下界，并证明分支孤立的不变曲线重数至多为叶状结构重数的两倍，最后将局部方法推广至复射影平面。
We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d),\ d\in\Bbb B$ minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on…
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…
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.
Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$
本文研究射影空间上一般奇异曲线全纯叶状结构，证明满测度集合中叶状结构的奇点均可线性化且双曲，并在次数至少为2时无不变代数曲线。
Length control for foliations by Jordan curves via smooth quasiconformal maps
针对单连通域上Jordan曲线叶状结构的长度控制问题，构造光滑5-拟共形映射，以ε=2/157肯定回答了Bouchala与Koskela提出的p=2及附近的可积性问题。
We consider a finite composition of generalized Hénon mappings $\mathfrak{f}:\mathbb{C}^2\to\mathbb{C}^2$ and its Green function $\mathfrak{g}^+:\mathbb{C}^2\to\mathbb{R}_{\ge 0}$ (see Section 2). It is well known that each level set $\{\mathfrak{g}^+=c\}$ for $c>0$ is foliated by biholomorphic images of $\mathbb{C}$ and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in $\mathbb{P}^2$ (see Section 4). Namely, for any injective holomorphic parametrization of any leaf, its derivative is bounded over $\mathbb{C}$ with respect to the Fubini-Study me…
Given a smooth foliation by complex curves (locally around a point $x\in\mathbb{C}^2\setminus\{0\}$) which is "compatible" with the foliation by spheres centered at the origin, we construct a smooth real-valued function $g$ in a neighborhood of said point, which is positive, homogeneous and constant along the leaves. A corollary we obtain from this is relevant to the problem of "bumping out" certain pseudoconvex domains in $\mathbb{C}^3$.
We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for th…
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 \[ M_4 (Γ) = \int_Ω\frac{|q|^2}{(l_Γ) ^4} \mathrm{d} L^3.\]
Le tissu dual d'un pré-feuilletage convexe réduit sur $\mathbb{P}^{2}_{\mathbb{C}}$ est plat
本文证明了复射影平面 $\mathbb{P}^{2}_{\mathbb{C}}$ 上约化凸预叶状结构的对偶织网是平坦的，推广了此前仅针对不变直线情形的结果。
In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting…
On the neighborhood of a torus leaf and dynamics of holomorphic foliations
本文研究复曲面中椭圆曲线邻域的 Ueda 分类与全纯叶状结构和乐动力学的关系，证明当存在沿某闭曲线的和乐为无理中性且不可线性化时该邻域为 ($\gamma$) 型，并基于 Pérez-Marco 的刺猬理论探讨相关线丛的度量半正性。
We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natu…
The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
本文使用 Loewner-Kufarev 方程研究黎曼球面的弦弧 Jordan 曲线叶状结构，证明了局部卷绕函数的 Dirichlet 能量与 Loewner 驱动测度的 Loewner-Kufarev 能量等价，并据此给出了 Weil-Petersson 拟圆周的新定量刻画。
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