A. Vidras, A. Yger
Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobi's Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity', in projective or toric situations.
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We present a restricted version of some affine Jacobi's residue formula (on an affine algebraic variety) with applications to higher dimensional (and affine) analogues of Wood's (or Reiss's) relations about the interpolation of pieces of analytic manifolds. We also compare algebraic and analytic constructions of residue currents along analytic cycles and sketch some of their main properties. This work extend previous work of the authors in the affine space.
Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formul…
A residue formula for integrals with hyperplane singularities
本文提出了沿仿射超平面奇点积分的新留数公式，借助受全正性启发的实矩阵稳定性概念，为相关积分计算提供了新工具。
In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues.
In previous work of the authors and their collaborators (see Progress in Math, vol. 114, Birkäuser, 1993) it was shown how the equivalence of several constructions of residue currents associated to complete intersection families of (germs of) holomorphic functions in ${\bf C}^n$ could be profitably used to solve algebraic problems like effective versions of the Nullstellensatz. In this work we explain how an application of similar ideas in the non-complete intersection case leads to a remarkable algebraic result, namely: Let $P_1,...,P_n$ be $n$ polynomials in $n$ variables such that the zero …
Toric Euler-Jacobi vanishing theorem and zeros at infinity
本文研究了托里安欧拉-雅各比消失定理与无穷远处零点存在的关系，通过几何条件和商环维数分析，建立了全局消失与零点存在的等价性。
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 prove residual formulas for vector fields defined on compact complex orbifolds with isolated singularities and give some applications of these on weighted projective spaces.
Given a finite group of automorphisms of a compact Riemann surface, the Chevalley-Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss-Manin connection.
Residue currents of cohesive modules and the generalized Poincaré-Lelong formula on complex manifolds
本文发展复流形上凝聚模的留数流理论，证明其对偶原理与比较公式，并据此建立适用于无全局局部自由分解的凝聚层的广义 Poincaré–Lelong 公式。
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.
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 study residue formulas for push-forward in K-theory of homogeneous spaces. First we review formulas for classical groups, which we derive from a formula for the classical Grassmannian case. Next we consider the homogeneous spaces for G2. One of them embeds in the Grassmannian Gr(2,7). We find its fundamental class in the equivariant K-theory and obtain the residue formula for the push-forward. This formula is valid for G2/B as well.
The Gelfond-Khovanskii residue formula computes the sum of the values of any Laurent polynomial over solutions of a system of Laurent polynomial equations whose Newton polytopes have sufficiently general relative position. We discuss two important consequences of this result: an explicit elimination algorithm for such systems and a new formula for the mixed volume. The integer coefficients that appear in the Gelfond-Khovanskii residue formula are geometric invariants that depend only on combinatorics of the polytopes. We explain how to compute them explicitly.
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。
We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincaré-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.
We present an explicit formula for computing toric residues as a quotient of two determinants, a la Macaulay, where the numerator is a minor of the denominator. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.
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 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 the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.
The parabolic Verlinde formula: iterated residues and wall-crossings
本文通过比较几何不变量理论中的穿墙现象与迭代留数泛函，对所有秩重新证明了抛物型 Verlinde 公式，并发展了 Hecke 对应的重言式变体、计算了模空间 Hilbert 多项式，给出了 rho-移位问题的新局部方法。
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.
The aim of this paper is to describe how to obtain residue-type formulas for push-forwards in equivariant cohomology, using the Jeffrey-Kirwan nonabelian localization theorem and the related result of Guillemin and Kalkman. This paper relates the description of equivariant push-forwards for homogeneous spaces of classical Lie groups with the action of the maximal torus with symplectic geometry, using the example of projective spaces and classical Grassmannians.
We show a residues formula for maps generically transversal to regular holomorphic distributions.
We give an algebraic description of wave fronts that appear in strictly hyperbolic Cauchy problem. Concrete form of definig function of wave front issued from initial algebraic variety is obtained by the aid of the Gauss-Manin systems satisfied by Gel'fand -Leray integrals.
The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties.
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。
We present a compact formula computing rational trigonometric sums. E. Verlinde's expression for the dimension of conformal blocks in WZW theory is an example of such a sum. As an application, we show that a formula of Bismut and Labourie for the Riemann-Roch numbers of moduli spaces of flat connections on a Riemann surface coincides with Verlinde's expression.
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.
We interpret the Combinatorial Nullstellensatz of Noga Alon as a multidimensional residue formula, describe some consequences of this interpretation and related open problems.
We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for comput…
For a smooth map between noetherian schemes, Verdier relates the top relative differentials of the map with the twisted inverse image functor `upper shriek'. We show that the associated traces for smooth proper maps can be rendered concrete by showing that the resulting theory of residues satisfy the residue formulas (R1)--(R10) in Hartshorne's "Residues and Duality". We show that the resulting abstract transitivity map relating the twisted image functors for the composite of two smooth maps satisfies an explicit formula involving differential forms. We also give explicit formulas for traces o…
Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle, can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases, can be expressed as special cases of one formula, involving the Weyl group action on the characters of the natural representation of the torus.
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。
For any polynomial mapping $F=(F_1,\dots ,F_n)$ of $\Cal C^n$ with a finite number of zeros we define the Noether exponent $ν(F)$. We prove the Jacobi formula for all polynomials of degree strictly less than $\sum_{i=1}^n (°F_i-1)-ν(F)$.
We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.
Quadratic residue patterns of length 4 and 5
该研究将二次剩余模式结果推广到特定字母表上长度为4和5的任意单词，通过低亏格超椭圆曲线的弗罗贝尼乌斯迹得到模式出现次数公式，并利用广义佐藤 - 泰特方法描述l = 4时归一化误差项的极限分布。
We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton polyhedron. In particular, up to an easy non-zero factor, this residue only depends on the (tau_0)-principal part of f, where tau_0 is the smallest face of the Newton polyhedron intersecting the diagonal. This formula allows us to prove some vanishing results for the residue. More precisely, we prove that the residue vanishes when tau_0 is unstable, and we give a…
The celebrated Smith-Minkowski-Siegel mass formula expresses the mass of a quadratic lattice (L, Q) as a product of local factors, called the local densities of (L,Q). This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly, by constructing a smooth integral group scheme model for an appropriate orthogonal group. Our method works for any unramified finite extension of Q_2. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generaliz…
Geometric Realization of Finite Residue Casimirs and Weil Operators via Szegő Kernels
本文针对有限域上带无穷远点的光滑射影曲线，通过Szegő核实现有限剩余卡西米尔与Weil算子的几何化，为相关配对研究提供了几何基础。
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