Marco Brunella
These are lecture notes of a course given in Pisa, SNS, in february 2002. They provide a classification of holomorphic foliations of nongeneral type on compact Kaehler surfaces.
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Let $\mathcal{F}$ denote a singular holomorphic foliation on $\mathbb{P}^2$ having a finite automorphism group $\mbox{aut}(\mathcal{F})$. Fixed the degree of $\mathcal{F}$, we determine the maximal value that $|\mbox{aut}(\mathcal{F})|$ can take and explicitly exhibit all the foliations attaining this maximal value. Furthermore, we classify the foliations with large but finite automorphism group.
In this work, we construct some irreducible components of the space of two-dimensional holomorphic foliations on $\mathbb{P}^n$ associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. We give a description of the generalized Kupka components, obtaining a classification of them in terms of the degree of the foliations, in both cases $n=3$ and $n=4$.
Given a (singular, codimension 1) holomorphic foliation F on a complex projective manifold X, we study the group PsAut(X, F) of pseudo-automorphisms of X which preserve F ; more precisely, we seek sufficient conditions for a finite index subgroup of PsAut(X, F) to fix all leaves of F. It turns out that if F admits a (possibly degenerate) transverse hyperbolic structure , then the property is satisfied; furthermore, in this setting we prove that all entire curves are algebraically degenerate. We prove the same result in the more general setting of transversely projective foliations, under the a…
Holomorphic foliations of degree four on the complex projective space
研究复射影空间上四次全纯叶状结构的结构分类，并对高阶射流的 $d\geq4$ 次叶状结构证明其必属于横截仿射、横截射影或有理拉回三种情形之一。
Contents 1. Algebraicity criterion: statement 2. Proof of the algebraicity criterion. 3. Pseudoeffectivity and movable classes. 4. Harder-Narasimhan filtrations and pseudo-effectivity. 5. Pseudo-effectivity of relative canonical bundles. 6. Rational curves and non-pseudoeffectivity of the canonical/cotangent bundles. 7. Birational stability of the cotangent bundle 8. Shafarevich-Viehweg conjecture. 9. Numerically trivial foliations. 10. The Beauville-Bogomolov-Yau decomposition 11. Some questions 11.1. Pseudoeffectivity of determinants. 11.2. Variations on Abundance. 11.3. Special manifolds. R…
Non-dicritical codimension one foliations on projective spaces of dimension four or higher always have an invariant algebraic hypersurface. The proof relies on a strengthening of a result by Rossi on the algebraization/continuation of analytic subvarieties of projective spaces.
Stability of pullbacks of foliations on weighted projective spaces
该研究证明了加权射影空间上带分裂切层的叶状结构的拉回所对应的射影空间叶状结构的稳定性定理，可构造多个新的叶状结构空间不可约分支，并为其他叶状结构族的稳定性提供统一替代证明。
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.
Following T. Suwa, we study unfoldings of algebraic foliations and their relationship with families of foliations, making focus on those unfoldings related to trivial families. The results obtained in the study of unfoldings are then applied to obtain information on the structure of foliations on projective spaces.
We prove that a transversely product component of the singular set of a holomorphic foliation on $\mathbb P^n$ is necessarily a Kupka component.
Transversely projective structures on smooth foliations on surfaces
本文在 Brunella 分类及 Biswas–Dumitrescu 关于奇异与正则横截射影结构差异的结果基础上，刻画了紧复曲面上不具有正则横截射影结构的光滑叶状结构。
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。
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 $\ge 3$.
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。
In this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes. We then specialize to holomorphic distributions and foliations. In particular, we provide sufficient conditions for distributions of arbitrary rank on projective spaces to be uniquely determined by their singular schemes.
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…
Unlikely intersections of codimension one foliations
该研究聚焦复射影流形上的奇异全纯叶状结构族，探究其整体相交所定义的余维数意外偏低的叶状结构，推进了复几何与代数几何领域的叶状结构相交理论研究。
We prove that, if $n\geq 3$, a singular foliation $\mathcal{F}$ on $\mathbb P^n$ which can be written as pull-back, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ of degree $d\geq2$ with one or three invariant lines in general position and $f:{\mathbb P^n}--->{\mathbb P^2}$, $deg(f)=ν\geq2,$ is an appropriated rational map, is stable under holomorphic deformations. As a consequence we conclude that the closure of the sets $\{\mathcal {F}= f^{*}(\mathcal{G})\}$ are new irreducible components of the space of holomorphic foliations of certain degrees.
We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\mathbb P^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.
This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of P^n to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional holomorphic foliation F of M is a morphism Φ:L -> TM where L is a holomorphic line bundle on M. The singular set of F is the analytic subvariety sing(F) = {p : Φ(p)=0} and the leaves of F are the leaves of the nonsingular foliation induced by F on M-sing(F). If M is P^n then, since line bundles over P^n are classified by the Chern class c_1(L) in H^2(P^n,Z) = …
We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is induced by a linear diagonal vector field.
We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable sense.
We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a com…
Homogeneous pre-foliations of co-degree one and degree four on the projective plane
本文通过射影自同构分类复平面上一阶和四阶均匀预叶形，其Legendre变换定义平坦四网，根据底层均匀叶形的类型区分不同情况，结合曲率-全纯性准则和符号计算得出有限列表。
We show that up to automorphisms of $\mathbb{P}^2_{\mathbb C}$ there are $5$ homogeneous convex foliations of degree four on $\mathbb{P}^2_{\mathbb C}.$ Using this result, we give a partial answer to a question posed in $2013$ by D. {Marín} and J. {Pereira} about the classification of reduced convex foliations on~$\mathbb{P}^2_{\mathbb C}.$
The space of codimension one holomorphic foliations of degree 1 in a projective space has an irreducible component whose general element is a logarithmic differential 1-form with simple poles in three hyperplanes. We compute its projective degree by resolving its rational parametrization map through succesive blow-ups with smooth centers.
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。
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。
Birational Geometry of Special Quotient Foliations and Chazy's Equations
本文研究了特殊商叶结构的理性几何，通过与恰兹方程的联系，提供了叶结构的显式模型及对称性分析。
We introduce and give normal forms for (one-dimensional) Riccati foliations (vector fields) on $\ov \bc \times \bc P(2)$ and $\ov \bc \times \ov \bc^n$. These are foliations are characterized by transversality with the generic fiber of the first projection and we prove they are conjugate {\em in some invariant Zariski open subset} to the suspension of a group of automorphisms of the fiber, $\bc P(2)$ or $\ov \bc^n$, this group called {\it global holonomy}. Our main result states that given a finitely generated subgroup $G$ of $\Aut(\bc P (2))$, there is a Riccati foliation on $\ov \bc \times \…
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
Singular Levi-flats and holomorphic foliations on projective spaces
本文研究射影空间上具有不变奇异Levi-平坦簇的余维1全纯叶状结构，引入两类条件刻画其可积性，证明Darboux型二择性并得到闭有理或对数1-形式的存在性。
Homogeneous Convex Foliations of degree 6
本文研究复射影平面上的齐次凸叶状结构，利用射影对偶性关联叶状结构与对偶平面$d$-网，分类了具三个径向奇点的齐次凸叶状结构并完成6次凸齐次叶状结构分类。
We show that up to automorphisms of $\mathbb P^2_{\mathbb C}$ there are $14$ homogeneous convex foliations of degree $5$ on $\mathbb P^2_{\mathbb C}.$ We establish some properties of the Fermat foliation $\mathcal F_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_H^{5}$ of degree $5.$ As a consequence, we obtain that every reduced convex foliation of degree $5$ on $\mathbb P^2_{\mathbb C}$ is linearly conjugated to one of the two foliations $\mathcal F_{0}^{5}$ or $\mathcal{F}_H^{5},$ which is a partial answer to a question posed in $2013$ by D. Mar\'ın and J.V. P…
Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.
Stability and Logarithmic Foliations on Complex Projective Spaces
本文在复射影空间上定义射影$L$-稳定性，证明其对极大不变代数除子与非临界非节点奇点条件可推出全局对数叶状结构，并应用于$\mathbb P^2$的拓扑刚性。
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…
Related to the classification of regular foliations in a complex algebraic surface, we address the problem of classifying the complex surfaces which admit a flat pencil of foliations. On this matter, a classification of flat pencils which admit foliations with a first integral of genus one and isolated singularities was done by Lins Neto. In this work, we complement Lins Neto's work, by obtaining the classification of compact complex surfaces which have a pencil with an invariant tangency set.
Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental p…
We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in the complement of a line. We establish the position of these foliations in the birational classification of foliations and prove the finiteness of their birational symmetries. Most of the results apply to wider classes of foliations.
We show that if $(X_t,\mathcal{F}_t)_t$ is a family of foliations with reduced singularities on a smooth family of surfaces, then invariance of plurigenera holds for sufficiently large $m$. On the other hand, we provide examples on which the result fails, for small values of $m$.
We study the homotopy groups of generic leaves of logarithmic foliations on complex projective manifolds. We exhibit a relation between the homotopy groups of a generic leaf and of the complement of the polar divisor of the logarithmic foliation.
Jets of flat partial connections
该研究在光滑叶状结构与局部自由层平坦偏联络的背景下定义并研究平坦偏联络的射流，分别刻画了余维1叶状结构的横截仿射、射影结构，还利用法层Bott联络的射流定义任意余维叶状结构横截射影结构的延拓并构造奇异结构。
The moduli space of algebraic foliations on P2 of a fixed degree and with a center singularity has many irreducible components. We find a basis of the Brieskorn module defined for a rational function and prove that set of pull-back foliations forms an irreducible component of the moduli space. The main tools are the Picard-Lefschetz theory of a rational function in two variables, period integrals, and the Brieskorn module.
Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension…
The set $\mathbf{F}(3)$ of foliations of degree three on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $23$ on which acts $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$. The subset $\mathbf{FP}(3)$ of $\mathbf{F}(3)$ consisting of foliations of $\mathbf{F}(3)$ with a flat Legendre transform (dual web) is a Zariski closed subset of $\mathbf{F}(3)$. We classify up to automorphism of $\mathbb{P}^{2}_{\mathbb{C}}$ the elements of $\mathbf{FP}(3)$. More precisely, we show that up to automorphism there are $16$ foliations of degree three wit…
Let $\mathcal{F}$ be written as $ f^{*}(\mathcal{G})$, where $\mathcal{G}$ is a $1$-dimensional foliation on $ {\mathbb P^{n-1}}$ and $f:{\mathbb P^n}--->{\mathbb P^{n-1}}$ a non-linear generic rational map. We use local stability results of singular holomorphic foliations, to prove that: if $n\geq 4$, a foliation $\mathcal{F}$ by complex surfaces on $\mathbb P^n$ is globally stable under holomorphic deformations. As a consequence, we obtain irreducible components for the space of two-dimensional foliations in $\mathbb P^n$. We present also a result which characterizes holomorphic foliations o…
We describe the topological types of leaves of generic logarithmic foliations on the complex projective plane. We prove that all leaves, except for a finite many are biholomorphic to $\mathbb{C}$ or homeomorphic to the surface known as Loch Ness monster.
We study conformal structure and topology of leaves of singular foliations by Riemann surfaces.
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 study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.
We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a foliation $\mathcal{F}_0$ on $\mathbb{C}^2$. We show that under some dynamical restriction on $\mathcal{F}_0$, we have two possibilities: $\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\{\mathcal{F}_t\}_{t \in \mathbb{D}_ε}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the…
We study local positive harmonic currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguyên says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.
A singular real analytic foliation $\mathcal{F}$ of real codimension one on an $n$-dimensional complex manifold $M$ is Levi-flat if each of its leaves is foliated by immersed complex manifolds of dimension $n-1$. These complex manifolds are leaves of a singular real analytic foliation $\mathcal{L}$ which is tangent to $\mathcal{F}$. In this article, we classify germs of Levi-flat foliations at $(\mathbb{C}^{n},0)$ under the hypothesis that $\mathcal{L}$ is a germ holomorphic foliation. Essentially, we prove that there are two possibilities for $\mathcal{L}$, from which the classification of $\…
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