Jean-Paul Brasselet, Maurício Corrêa, Fernando Lourenço
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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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…
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。
Baum-Bott residue of flags of holomorphic distributions
本文将残数理论从全纯叶层旗扩展至全纯分布旗，提供了有效计算方法，并推导出 $\mathbb{P}^{3}$ 上分布旗中分布次数、切向阶、欧拉示性数与曲线次数间的关系。
We prove inequalities relating the degrees of holomorphic distributions and of holomorphic foliations forming a flag on $\mathbb{P}^n$. Such inequalities are inspired by the so called Poincaré problem for foliations.
In this work we prove a Baum-Bott type formula for non-compact complex manifold of the form $\tilde{X}=X- \mathcal{D}$, where $X$ is a complex compact manifold and $\mathcal{D}$ is a normal crossing divisor on $X$. As applications, we provide a Poincaré-Hopf type Theorem and an optimal description for a smooth hypersurface $\mathcal{D}$ invariant by an one-dimensional foliation $\mathscr F$ on $\mathbb{P}^n$ satisfying $Sing(\mathscr F) \subsetneq \mathcal{D}.$
This is a survey paper dealing with holomorphic foliations, with emphasis on residue theory and its applications. We start recalling the definition of holomorphic foliations as a subsheaf of the tangent sheaf of a manifold. The theory of Characteristic Classes of vector bundles is approached from this perspective. We define Chern classes of holomorphic foliations using the Chern-Weil theory and we remark that the Baum-Bott residue is a great tool that help us to classify some foliations. We present throughout the survey several recent results and advances in residue theory. We finish by presen…
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。
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。
In this work we prove a Brunella-Khanedani-Suwa variational type residue theorem for currents invariant by holomorphic foliations. As a consequence, we give conditions for the leaves of a singular holomorphic foliation to accumulate in the intersection of the singular set of the foliation with the support of an invariant current.
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。
The Schwartz index and the residue of logarithmic foliations along a hypersurface with isolated singularities
本文证明了对数叶状结构沿孤立奇点超曲面的Baum-Bott型公式，用Schwartz指数表达留数，并推广了Poincaré问题的障碍结果至任意偶数维射影空间。
Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella
该综述性文章研究向量场与全纯叶状结构的不变量，旨在关联复解析奇异簇与复流形上奇异全纯叶状结构这两个有诸多共通点的理论领域。
Inequalities and enumerative formulas for flags of Pfaff systems
本文研究了Pfaff系统在复射影空间上的旗帜不等式与计数公式，推导了与度数相关的不等式，并证明了某些全纯可积分布的稳定性结果。
Residue formula for flag manifold of type $A$ from wall-crossing
本文考虑 $A$ 型旗流形上的等变积分，利用受 Takuro Mochizuki 穿墙公式启发的计算方法，重新证明了 Weber 和 Zielenkiewicz 的留数公式，并通过设置 $K$ 理论类导出了 Grothendieck 多项式的行列式公式。
Formulas for Residues of Type Camacho-Sad and Applications
本文给出了全纯叶状结构关于不变解析子簇的Camacho-Sad型留数之和的公式，并将其应用于射影叶状结构，建立了留数与不变子簇特征的联系，同时给出了非孤立奇点超曲面的邻接公式与Milnor数显式公式。
We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree $d$.
The paper consist of two parts. In the first part we introduce flags of lattices and associated injective systems of (non-normal) cones and projective systems of (non-normal) affine toric varieties. We study the associated field of multidimensional Laurent power series. In the second part we use the resolution of singularities techniques to study the geometry near a complete flag of subvarieties and the Parshin's residues. The first part plays the role of a standard coordinate neighborhood for Parshin's points.
On Bott's residue formula for toric complete intersections
本文研究了紧致纹理轨道上完全交集的博特残差公式，通过分析正则分布和建立庞加莱型不等式，探讨了分布奇点数目的计算方法及其应用。
We study a notion of derived foliations on schemes and derived schemes of arbitrary characteristics. We introduce the Hodge filtration associated to a derived foliation, which functorialy filters derived de Rham cohomology. We use this filtration to study vanishing results of Chern classes of perfect complexes endowed with connexions along derived foliations. As an application, we prove a positive characteristic version of Bott's vanishing theorem and more generally of existence of residues for foliations with singularities due to Baum-Bott in characteristic zero.
The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P^3. We construct a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.
The purpose of this thesis is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P3. I construct a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.
Baum-Bott residue currents
该研究在复流形全纯叶状结构的奇点紧分支上，构造了表示Baum-Bott留数的显式留数电流，分析了其与度量联络选取的无关性，并在秩1及孤立奇点情形下给出对应描述。
On functoriality of Baum-Bott residues
该研究证明了Baum-Bott残差在特定条件下的函子性，并解决了多个关于叶形和Poisson结构的长期问题。
Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool, similar to generating sequences, which we call the iterated residue technique. We prove new formulas on the calculus of iterated residues and use them to prove straightening laws and multiplication formulas for stable Grothendieck polynomials. As a further application of our method, we give new proofs that the $K$-Pieri rule and the expansions of Grothendieck po…
Global Chern currents of coherent sheaves and Baum Bott currents
本文将凝聚解析层特征类与全纯叶状结构Baum-Bott残数的表示扩展至任意复流形，结合Green的Chern类定义构造了支撑于特定集合上的代表元，并证明了新的调配公式。
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