Oliver Klehn
We define local residues of holomorphic 1-forms on an isolated surface singularity that have isolated zeros and prove that a certain residue equals the index of the 1-forms.
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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。
We investigate the relative logarithmic connections on a holomorphic vector bundle over a complex analytic family. We give a sufficient condition for the existence of a relative logarithmic connection on a holomorphic vector bundle singular over a relative simple normal crossing divisor. We define the relative residue of relative logarithmic connection and express relative Chern classes of a holomorphic vector bundle in terms of relative residues.
We develop a new approach to the study of the multipoint loci of holomorphic maps between complex manifolds. We relate the $k$-fold locus to the curvilinear component of the Hilbert scheme of $k$ points on the source space of the map, and using equivariant localisation, we derive a closed iterated multipoint residue formula. Our work is motivated by ideas and conjectures of M. Kazarian and R. Rimányi on residual polynomials.
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 .
We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.
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 study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.
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.
We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S. A normal surface X with isolated cyclic quotient singularities {S_i} admits a Q-Gorenstein partial smoothing to a surface with singularities given by the residues of the S_i. We define the singularity content of a Fano lattice polygon P: this records the total number of elementary T-singularities and the residues of the corresponding toric Fano surface X_P. We express the degree of X_…
A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset $S$ of $X$, and fix an endomorphism $A(x) \in \text{End}(E_x)$ for every $x \in S$. It is natural to ask when there is a logarithmic connection on $E$ singular over $S$ with residue $A(x)$ at every $x \in S$. We give a necessary and sufficient condition for it under the assumption that the residues $A(x)$ are rigid.
The periodic cyclic homology of any proper dg category comes equipped with a canonical pairing. We show that in the case of the dg category of matrix factorizations of an isolated singularity the canonical pairing can be identified with the so-called higher residue pairing on the twisted de Rham cohomology of the singularity.
On residue localization for complex supermanifolds
针对费米子和玻色子维数相等的紧复超流形，证明了奇全纯向量场的全纯留数局部化公式，并在约化零点集孤立非退化假设下给出显式局部留数公式。
The index of a real vector field at an isolated complete intersection singularity
本文将实解析超曲面上孤立奇点处向量场指标公式推广到完全交奇点情形，利用广义 Koszul 复形上的广义留数符号建立双线性型演算，从而用符号差表达指标。
We show a residues formula for maps generically transversal to regular holomorphic distributions.
Residue functions and Extension problems
针对Demailly定性延拓定理缺乏延拓估计的问题，研究利用残差函数获取一般情形下全纯延拓的$L^2$估计的可能性，并证明指标1残差函数定义的1-lc测度等于Ohsawa测度。
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。
Residues and Gorenstein Contractions of Genus One Curves
本文研究了局部艺术代数环上亏格为一的节点曲线的余数及其在热带数据下的推广，进而构造出将子曲线收缩为格森亏格为一奇点的收缩映射。
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。
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational con…
Let a finite abelian group $G$ act (linearly) on the space $\mathbb{R}^n$ and thus on its complexification $\mathbb{C}^n$. Let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in general $W \neq \mathbb{R}^n/G$). We give an algebraic formula for the radial index of a 1-form on the real quotient $W$. It is shown that this index is equal to the signature of the restriction of the residue pairing to the $G$-invariant part $Ω^G_ω$ of $Ω_ω= Ω^n_{\mathbb{R}^n,0}/ω\wedge Ω^{n-1}_{\mathbb{R}^n,0}$. For a $G$-invariant function $f$, one has the so-called quantum cohomology group defined in the qu…
Generic vector fields on isolated complex hypersurface germs
本文研究孤立超曲面奇点上的全纯向量场，推导出全纯向量场存在的全局障碍。通过GSV指数刻画向量场的最小可能指数，并证明其与Tjurina-Greuel数的关系。进一步证明当向量场可延拓到C^{n+1}时，指数相等，且此类延拓构成开密集集。
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。
Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic …
An old theorem of Weil and Kodaira says that for a compact Kähler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this paper, we generalize the above theorem to general compact complex manifolds by showing that the necessary and sufficient condition in general is described by a holomorphic invariant called the $\mathcal Q$-flat class. Next, we prove that the holomorphic criterion is reduced to the topological one when $X$ has Property $(H)$. Since all Kähler manifolds have Pro…
Consider a normal complex analytic surface singularity. It is called Gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically Gorenstein if this line bundle is topologically trivial. The second notion depends only on the topological type of the singularity. Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic topological type. The question to know if any n…
Hironaka's characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition $ (\mathrm{Pol}) $ holds, or a mild condition $(*) $ on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.
Suppose $(M,ω)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $μ: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = μ^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant Kähler structure. In an earlier paper, we proved a formula (the residue formula) for $η_0 e^{ω_0}[M_{red}]$ for any $η_0 \in H^*(M_{red})$, where $ω_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $η_0 = Td(M_{red})$; when $K$ acts freely on $μ^{-1}(0)$ this yields a formula for the Rie…
We consider a holomorphic 1-form $ω$ with an isolated zero on an isolated complete intersection singularity $(V,0)$. We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair $(V,ω)$. They generalize the Eisenbud-Levine-Khimshiashvili quadratic form defined for a smooth $V$.
Suppose $M^{n+1}$ is a complex manifold and K is a hypersurface with isolated singularities. Let $ω$ be a holomorphic form on $M\setminus K$ with the first order pole on K. The Leray residue of such form gives an element in the n-th homology of K which is the Alexander dual to $[ω]\in H^{n+1}(M\setminus K)$. It always lifts to the intersection homology if 0 does not belong to the spectra of a singular points. We assume that singularities are described by the quasihomogeneous equations in certain coordinate systems. Suppose that oscillation indicators of the singular points are greater then -1.…
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。
We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.
Following the work of Daniel Barlet ([Bar97]) and Ridha Belgrade ([Bel01]) the aim of this article is the study of the existence of $(a, b)$-hermitian forms on regular $(a, b)$-modules. We show that every regular $(a,b)$-module with a non-degenerate bilinear form can be written in an unique way as a direct sum of $(a, b)$-modules $E_i$ that admit either an $(a, b)$-hermitian or an $(a, b)$-anti-hermitian form or both; all three cases are equally possible with explicit examples. As an application we extend the result in [Bel01] on the existence for all $(a, b)$-modules $E$ associated with the B…
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence corresponding to the anti-holomorphic filtration of this algebra and compute its E1 term. This spectral sequence converges to the L2 cohomology H2*(U, V). We show that this DGLA is formal, although it does not always satisfy to the d'd''-Lemma.
We define the counting of holomorphic cylinders in log Calabi-Yau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focus-focus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross…
Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. (i) A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. T…
Let X be a complex analytic space. A short analytic arc is a holomorphic map of the closed unit disc to X such that only the origin is mapped to a singular point. In contrast with the space of formal arcs studied by Nash, the moduli space of short analytic arcs usually has infinitely many connected components. We describe these for surface singularities, in terms of certain conjugacy classes of the fundamental group of the link. For quotient singularities (in any dimension), this gives a concrete realization of the McKay correspondence. Our results also give new connections between a surface c…
We study the local holomorphic Euler characteristic $χ(x,\mathcal{F})$ of sheaves near a surface singularity obtained from contracting a line $\ell$ inside a smooth surface $Z$. We prove non-existence of sheaves with certain prescribed numerical invariants. Non-existence of instantons on $Z$ with certain charges follows, and we conclude that $\ell^2$ poses an obstruction to instanton decay. A Macaulay 2 algorithm to compute $χ$ is made available at http://www.maths.ed.ac.uk/~s0571100/Instanton/
We formulate certain sufficient conditions for the symplectic monodromy of an isolated quasihomogeneous singularity to be of infinite order in the relative symplectic mapping class group of the Milnor fibre and give a proof using Maslov classes, stability theory for Lagrangian folds resp. stable Morse theory for generating families as well as algebraic results about relative cohomology of smoothings of isolated singularities. Our conditions being slightly more restrictive than Seidel's, in contrary to Seidel's proof, we do not use Floer theory to derive this result. An alternative approach usi…
A Poisson--Poincaré--Dulac for Poisson Connections
研究具有对数极点的泊松平坦连接，在非共振条件下证明其可规范等价于具有常数交换留数的纯Euler–泊松标准型，并建立泊松设置下的对数Riemann–Hilbert对应。
Resolutions of differential operators of low order for an isolated hypersurface singularity
本文提出新方法研究孤立奇点分次超曲面环的微分算子，构造二阶和三阶微分算子模的极小生成集及自由解消，推广了已有结果，并利用奇点范畴中的矩阵分解实现构造。
Gorenstein singularities with $\mathbb{G}_m$-action and moduli spaces of holomorphic differentials
本文通过测试配置构造Gorenstein奇点，分解全纯微分的strata，并研究其变形空间性质，同时探讨非变strata的紧致化及$K(π,1)$性质。
The Bruce-Roberts number of holomorphic 1-forms along complex analytic varieties
本文定义全纯1-形式沿复解析簇的Bruce-Roberts数，证明其可用Ebeling--Gusein-Zade指标、Milnor数和Tjurina数表示，并应用于复二维全纯叶状结构。
A contact singularity is a normal singularity $(V,0)$ together with a holomorphic contact form $η$ on $V\backslash$ Sing $V$ in a neighbourhood of 0, i.e. $η\wedge (dη)^r$ has no zero, where dim $V=2r+1$. The main result of this paper is that there are no isolated contact singularities.
In this short note we would like to show how it is possible to use techniques introduced in the theory of local dynamics of holomorphic germs tangent to the identity to study global meromorphic self-maps of the complex projective space. In particular we shall show how a meromorphic self-map of a complex projective space induces a holomorphic foliation of the projective space in Riemann surfaces, whose singular points are exactly the fixed points and the indeterminacy points of the map; and we shall prove three index theorems, relating suitably defined local residues at the fixed and indetermin…
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。
K-differentials with prescribed singularities
该研究探讨了黎曼曲面上k≥3的亚纯k-微分的局部不变量，证明了给定零点阶数模式下本原全纯k-微分的存在性，并揭示了不同亏格下k-留数配置的出现规律。
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