Jets of singular foliations
Eduardo Esteves
给定一个奇异的叶理满足局部处处弗罗贝纽斯条件，即使在奇点，我们展示了如何构建其全球层的射流。我们的结构是纯形式的，因此适用于各种上下文。
Given a singular foliation satisfying locally everywhere the Frobenius condition, even at the singularities, we show how to construct its global sheaves of jets. Our construction is purely formal, and thus applicable in a variety of contexts.
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Jets of flat partial connections II
本文研究关于奇异叶状结构的平坦偏联络的射流，利用第一横截射流层解决偏联络向亚纯联络的扩张问题，并将其应用于横截仿射结构与不变超曲面等叶状结构理论经典问题。
On the Singular Scheme of Split Foliations
证明了余维1局部自由分布的切层分裂为线丛和的充要条件是其奇异概型为算术Cohen-Macaulay。此外，我们还证明了曲线叶理是由余维为1的一般横截全纯分布的交集给出的当且仅当其奇异方案是算术Buchsbaum。最后，我们证明了这些叶理是由它们的奇异格式决定的，并证明了余维$2$的某些算术Buchsbaum格式的Hilbert格式是对Grassmannian的双有理格式.
This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on projective manifolds.
Remarks on the second Chern class of a foliation
本文研究了叶状结构的第二陈类及其奇点方案的度数，提出对于射影空间上的叶状结构，其奇点方案的度数下界为d+1，并探讨了等号成立的条件。
Foliations by Curves with Curves as Singularities
设$\mathcal F$是$\mathbb{P}^n$上的一个全纯一维叶理，使得它的奇异轨迹$Σ$的分量是曲线$C_i$和点$p_j$ .我们确定的$p_j$的数量，计数的多重性，在不变量的$\mathcal F$和$C_i$，假设$\mathcal F$是特殊的沿$C_i$。只允许一个非零维分量的$Σ$，我们还证明了结果时，叶理恰好是由其奇异轨迹。
Singular locus of q-logarithmic foliations
该数学代数几何领域的研究明确了一般q-余维对数叶状结构的奇异轨迹结构及其与开折的关系，并在射影空间Pⁿ情形下算出了持续奇点概形对应的分次理想。
Bounds and formulas for residues of singular holomorphic foliations and applications
本文研究了一维全纯叶状结构的GSV指标公式及其在射影空间上Poincaré问题的应用。
In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini - Spicer and Spicer - Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective threefolds.
In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some conditions the index i of a Q-Fano foliation is bounded by the rank r of F, and classify the cases in which i=r. Next we consider Q-Fano foliations F for which i=r-1. These are called del Pezzo foliations. We classify codimension 1 del Pezzo foliations on mildly singular varieties.
In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context.
In this paper, we classify the configurations of the singular points which appear on the quotients of the projective plane by the $1$-foliations of degree $-1$ in characteristic $2$.
On the stability of foliations of degree 3 with a unique singular point
本文应用几何不变量理论研究P^2上具有唯一奇点且Milnor数为13的3阶叶状结构的稳定性，并针对重数2的情况分类刻画了稳定、严格半稳定和不稳定状态。
Projective Foliations with a Unique Singular Point and Maximum Algebraic Multiplicity
本文研究复射影平面上具有唯一奇点的余维一全纯叶状结构，聚焦次数为$d$且奇点最大重数为$d$的叶状结构，明确描述该类结构、刻画具唯一奇点的叶状结构，并给出其希尔伯特-芒福德稳定性的相关结果。
Stability of Holomorphic Foliations with Split Tangent Sheaf
证明了具有分裂切层且具有好奇异集的射影空间的奇异全纯叶理集在全纯叶理空间中是开的。作为应用，我们推广了Camacho-Lins Neto关于线性拉回叶理的一个结果，给出了余维$k \ge 2$的$\mathcal L$ -叶理的刚性的一个判别准则，并证明了Cerveau-Litti关于$\mathbb P^4$的余维1的$\mathcal L$ -叶理的刚性的一个猜想.这些结果使我们能够展示一些以前未知的奇异全纯叶理空间的不可约分量。
In this work we classify foliations on $\mathbb{CP}^3$ of codimension 1 and degree $2$ that have a line as singular set. To achieve this, we do a complete description of the components. We prove that the boundary of the exceptional component has only 3 foliations up to change of coordinates, and this boundary is contained in a logarithmic component. Finally we construct examples of foliations on $\mathbb{CP}^3$ of codimension 1 and degree $s \geq 3$ that have a line as singular set and such that they form a family with a rational first integral of degree $s+1$ or they are logarithmic foliation…
Numerically trivial foliations
给定复Kaehler流形上伪有效线丛的正奇异Hermite度量，构造了满足一定数值条件解析类比的奇异叶理.这个叶理细化了Tsuji的数值上微不足道的纤维化和Iitaka纤维化。利用伪有效线丛上具有解析奇点的几乎正奇异厄米特度量，构造了一种改进nef纤维化的叶状结构。如果叶理的奇点是孤立点，则叶理的余维数是线丛数值维数的一个上界，叶理可以解释为nef和Kodaira-Iitaka维数偏离的一个几何原因.对几个曲面实例进行了较详细的研究，其中$\mathbb{P}^2$的9点分解给出了叶子的数维数与余维数相等的反例。
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。
On non-Kupka points of codimension one foliations on $\mathbb{P}^3$
研究了$\mathbb{P}^3$上余维为1的全纯叶理的奇异集。我们找到了一个局部正规形的余维两个组件的奇异集，是不是Kupka型。我们还确定了$\mathbb{P}^3$上余维1叶理的奇异集的余维2分量中的非Kupka点的个数。
This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety $X$ is formally integrable at any point, and, if we suppose that its singular locus has codimension $\geq 2$, then the truncation of its analytification is a locally inte…
Radial foliations in dimension three
本文针对三维全纯叶状结构，定义近径向与径向叶状空间，证明径向叶状对应“开书”情形，并讨论一般近径向情形。
Invariant theory of foliations of the projective plane
研究了射影平面上奇异叶理的不变量理论。我们的第一个主要结果是：一个m>1次的叶理是不稳定的，仅当它在1维上有奇点或包含一个重数至少为（m^2-1）/（2 m +1）的孤立奇点.我们的第二个主要结果是构造了一个从m次叶理空间到m^2+m-2次曲线空间的不变映射。我们在m=2的情况下明确地描述这个映射。
Foliations with isolated singularities on Hirzebruch surfaces
本文研究Hirzebruch曲面上叶结构的奇点性质，证明在特定条件下叶结构由其奇点方案决定，并探讨了不同δ值下的不同结论。
On the singular scheme of codimension one holomorphic foliations in P^3
在这项工作中，我们首先表明，一个全纯叶理与奇异约化当且仅当其正常层是无挠的。此外，当奇异轨迹的余维数至少为2时，证明了约化等价于切层的自反性。我们的主要结果表明，一方面，P^3中余维为1的叶理的切层是局部自由的当且仅当奇异概型是一条曲线，并且它分裂当且仅当该曲线是算术Cohen-Macaulay。另一方面，我们讨论了P^3中的分裂叶理何时由它的奇异概型决定。
Regular Leaves of Singular Foliations
本文识别了一类奇异代数叶状结构，其通过奇异点的叶保持正则性，并利用剩余gerbes的存在性证明了相关定理，推广了Cerveau定理并给出了Zariski--Lipman猜想的新证明。
We will work with codimension one holomorphic foliations over the complex projective space, represented by integrable forms $ω\in H^0(Ω^1_{\PP^n}(e))$. Our main result is that, under suitable hypotheses, the Kupka set of the singular locus of $ω\in H^0(Ω^1_{\PP^3}(e))$, defined algebraically as a scheme, turns out to be arithmetically Cohen-Macaulay. As a consequence, we prove the connectedness of the Kupka set in $\PP^n$, and the splitting of the tangent sheaf of the foliation, provided that it is locally free.
Dynamics of Singular Holomorphic Foliations on the Complex Projective Plane
这份手稿是介绍复射影平面上的全纯叶理理论。从历史上看，这个问题已经出现在理论的常微分方程在复杂的领域和各种尝试解决希尔伯特的第16问题，但随着引进复杂的代数几何，叶理理论和动力系统，它现在已经成为一个有趣的主题。
Global residue formula for logarithmic indices of foliations
本文研究一维全纯叶状结构的对数指标，建立全局残差公式及奇点总和公式，并应用于复射影空间与奇异正规射影簇。
The Valuative Theory of Foliations
本文描述了复平面中（奇异）全纯叶理的无穷近奇点的赋值。他们似乎是那些满足L'Hopital规则的推广。利用它们，我们刻画了双临界向量场，一般简单奇点，非收敛解的存在性等。
Foliations with small singular set in arbitrary characteristic
研究任意特征下光滑代数簇上叶状结构几何性质，给出极小有理和德尔佩佐曲面上正则叶状结构分类，推广卡马乔 - 萨德指标，证明射影空间叶状结构相关结论，建立$p$-闭叶状结构的博特消没定理版本。
On the connectedness of the singular set of holomorphic foliations
本文研究全纯叶形奇点集的连通性，证明其在特定条件下必连通，并由此得到关于全纯分布可积性的拓扑障碍，回答了Cerveau关于三维射影空间上一维叶形的问题。
In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$ with some extra degree assumptions. We determine new irreducible components of the moduli space of codimension two singular projective foliations of any degree, and we show that they are generically reduced in their natural scheme structure. Our method is based on an explicit description of the Zariski tangent space of the corresponding moduli space at a given…
A codimension one singular holomorphic foliation is Newton non-degenerate if it satisfies the classical conditions of Kouchnirenko and Oka, in terms of its Newton polyhedra system. In this paper we prove that a foliation is Newton non-degenerate if and only if it admits a logarithmic reduction of singularities of a combinatorial nature.
Bounds on leaves of foliations of the plane
本文给出了复射影平面上（奇异代数）叶理的（广义代数）叶的次数的界的Poincare问题的解。第一个定理给出了一种新的界限，它涉及到叶的奇异轨迹的Castelnuovo-Mumford正则性。第二个定理给出了一个界的两个奇点数的叶子：总Tjurina数，和非准齐次奇点的数量。如果存在这样的奇点，那么这个界至少在曲线不可约时改进了杜普莱西斯和沃尔的界。
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…
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