Quy Thuong Lê, Khanh Hung Nguyen
We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.
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The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.
We consider families of Calabi-Yau n-folds containing singular fibres and study relations between the occurring singularity structure and the decomposition of the local Weil zeta-function. For 1-parameter families, this provides new insights into the combinatorial structure of the strong equivalence classes arising in the Candelas - de la Ossa - Rodrigues-Villegas approach for computing the zeta-function. This can also be extended to families with more parameters as is explored in several examples, where the singularity analysis provides correct predictions for the changes of degree in the dec…
Denef-Loeser zeta functions of suspensions and Lê-Yomdin singularities
本文证明了平面曲线奇点悬垂的解析猜想及Le-Yomdin表面奇点的解析与monodromy猜想，推导了悬垂超曲面的motivic和topological zeta函数公式。
Motivic Hilbert zeta functions for curve singularities and related invariants
本文证明了曲线奇点的motivic Hilbert zeta function可生成点状Hilbert概形的欧拉数，并揭示了其与半模生成函数、HOMFLY多项式及Severi层次数等不变量的关系，同时推广了Kawai关于奇异曲线欧拉数的结果。
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f in Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of the complex zero locus of f close to the origin. Essentially the entire paper is…
Singularities of symmetric powers and irrationality of motivic zeta functions
本文证明若簇具有L-有理奇点则其对称幂亦然，并由此将Kapranov动机zeta函数有理性蕴含的Kodaira维数负性与无偶数次微分形式这一结论从曲面推广到任意维数。
On a suspension formula for Denef-Loeser zeta functions
本文将Artal等人关于两点悬挂的拓扑zeta函数公式推广到任意悬挂的motivic层面，并进一步得到超孤立、Lê-Yomdin及加权Lê-Yomdin奇点相关非孤立奇点族的motivic zeta函数公式。
The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities, introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties, and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the Minimal Model Program, we introduce very general stringy invariants, associated to 'almost all' singularities, more precisely to all singularities which are not strictly log canonical. They specialize to the …
Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring O_{P,X} at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if O_{P,X} is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincare series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the gene…
The local motivic monodromy conjecture for simplicial nondegenerate singularities
本文证明了针对非退化奇点的局部动机monodromy猜想，并展示了其与局部拓扑zeta函数和Igusa zeta函数极点之间的联系。
Algorithm for motivic Hilbert zeta function of some curve singularities
本文提出算法计算特定曲线奇点的两种动机Hilbertζ函数，并提供Python实现。
In a previous work we have introduced the notion of embedded $\Q$-resolution, which essentially consists in allowing the final ambient space to contain abelian quotient singularities. Here we give a generalization of N. A'Campo's formula for the monodromy zeta function of a singularity in this setting. Some examples of its applications are shown.
Birational zeta functions
该研究定义了既约多项式动机ζ函数的双有理类似物，其具有弧接触轨迹的内蕴含义，并证明局部平面曲线奇点下该双有理ζ函数与动机ζ函数的极点本质一致。
We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to…
The Poincare series of an irreducible plane curve singularity equals the zeta function of its monodromy, by a result of Campillo, Delgado and Gusein-Zade. We derive this fact from a formula of Ebeling and Gusein-Zade relating the Poincare series of a quasi-homogeneous complete intersection singularity to the Saito dual of a product of zeta functions.
On Motivic Zeta Functions and Stringy E-function via Embedded $\mathbb{Q}$-Resolution
该研究通过嵌入$\u211a$-消解方法，推导半拟齐次奇点的动机Zeta函数公式及二维极点，提出弦E函数的新公式并用于半拟齐次、非退化多项式的弦E函数计算。
The auto-Igusa zeta function of a plane curve singularity is rational
研究证明平面曲线奇点芽的自 Igusa 函数具有有理性，并由此给出特征零代数闭域上平面曲线在一点光滑的新判据。
Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions
本文研究固定生成元数的单分支平面曲线 Hilbert 概形的动机类，给出公式、正性与拓扑不变性，并联系 Igusa zeta 函数。
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of thr…
The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For nondegenerate functions it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerat…
The $p$-adic zeta function of a plane curve singularity
本研究借助环面修改与相容性，在解析变量替换公式下计算平面曲线奇点的局部 $p$-进 zeta 函数，规避了代数情形与 Denef 公式。
In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to th…
In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces.
We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.
We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. Some previous invariants are no longer invariants for this new relation, however, thanks to a Denef & Loeser formula coming from motivic integration in a Nash setting, we managed to derive new invariants for this equivalence relation.
In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $Δ_l(t)$, and the relation between the monodromy zeta function and the Hodge spectrum of the singularity. We go further with $Δ_1(t)$ in the case $n=2$. This work is based on knowledge of multiplier ideals and local systems.
Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions
本文通过构造普通爆破算法将二元$C^{\infty}$函数零点簇化为几乎法向交叉形式，并借助van der Corput型引理，用零点簇的重数刻画了相应局部zeta函数的亚纯延拓区域。
We show some examples of topological zeta functions associated to an isolated plane curve singular point and an allowed, in the sense of Némethi and Veys, differential form that have several poles of order two. This is in contrast to the case of the standard differential form where, as showed by Veys for plane curves and by Nicaise and Xu in general, there is always at most one pole of order two.
Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings
本研究证明$K$-平凡簇的动机zeta函数及相关单值性猜想具有双有理不变性，扩展了已知适用范围，并揭示该函数可作为簇在$t=0$处光滑填充存在性的障碍。
We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the …
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize …
The aim of this paper is to describe explicitly the poles of the meromorphic continuation of the Igusa local zeta function associated to several polynomials. Using resolution of singularities is possible to express the Igusa's local zeta function as a finite sum of p-adic monomial integrals. We compute these monomial integrals using techniques of toroidal geometry. In this way, we obtain an explicit list for the candidates to poles of the local zeta function associated to several polynomials.
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
本文推广p进McKay对应，将Denef的Igusa局部zeta函数公式拓展至Q-嵌入奇点消解情形，证明了相关强单值群猜想与指数和猜想对特定加权齐次多项式成立。
Archimedean zeta functions, singularities, and Hodge theory
利用Hodge理论研究全纯函数f的阿基米德zeta函数Z_f的极点与奇点不变量之间的关系，证明Z_f的最大非平凡极点为f的最小指数的相反数，其阶由Bernstein-Sato多项式b_f(s)对应根的重数决定，并给出b_f(s)的根不是Z_f的极点的反例。
We prove that if two plane curve singularities are equisingular, then they are topologically equivalent. The method we will use is P.~Fortuny~Ayuso's who proved this result for irreducible plane curve singularities.
To a given analytic function germ $f:(\mathbb{R}^d,0) \to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in \mathbb{Z} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of B…
We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. Then we classify parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.
A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced in [GZLM1]. In this article we define the topological zeta function for meromorphic germs and we study its poles in the plane case. We show that the poles do not behave as in the holomorphic case but still do satisfy a generalization of the monodromy conjecture.
We present an intersection-theoretical approach to the invariants of plane curve singularities $μ$, $δ$, $r$ related by the Milnor formula $2δ=μ+r-1$. Using Newton transformations we give formulae for $μ$, $δ$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities.
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically equivalent.
The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta functi…
In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function Z(L,s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolution of singularities, and the main result is that Z(L,s) does not depend on this choice and depends only on L. Known properties of the topological zeta function pro…
We show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. In Theorem 1.1 we list integers can be attained as Milnor numbers of a given semi-quasi-homogeneous singularity.
Minimal compactifications of the affine plane with only star-shaped singularities
该研究针对复仿射平面的极小紧化问题，在已有对数典范奇点相关分类基础上，完成仅含星形奇点的仿射平面极小紧化的分类工作。
Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched …
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
We study the poles and residues of the complex zeta function $ f^s $ of a plane curve. We prove that most non-rupture divisors do not contribute to poles of $ f^s $ or roots of the Bernstein-Sato polynomial $ b_f(s) $ of $ f $. For plane branches we give an optimal set of candidates for the poles of $ f^s $ from the rupture divisors and the characteristic sequence of $ f $. We prove that for generic plane branches $ f_{gen} $ all the candidates are poles of $ f_{gen}^s $. As a consequence, we prove Yano's conjecture for any number of characteristic exponents if the eigenvalues of the monodromy…
If the first Betti number of the Milnor fibre of a plane curve singularity is zero, then the defining function is equivalent to $x^r$.
We characterize plane curve germes non-degenerate in Kouchnirenko's sense in terms of characteristics and intersection multiplicities of branches.
Motivic classes of curvilinear Hilbert schemes and Igusa zeta functions
本文研究了奇异簇的曲线型 Hilbert 概形，引入了基于奇点嵌入消解的 Igusa zeta 函数，并提出了计算其 motive 类的递归公式，同时分析了平面曲线奇点背景下的几何与组合学。
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.
We define a new equivariant (with respect to a finite group $G$ action) version of the Poincaré series of a multi-index filtration as an element of the power series ring ${\widetilde{A}}(G)[[t_1, \ldots, t_r]]$ for a certain modification ${\widetilde{A}}(G)$ of the Burnside ring of the group $G$. We give a formula for this Poincaré series of a collection of plane valuations in terms of a $G$-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve s…
We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation zeta functions of the special linear groups over the ring of integers are bounded above by 2. In order to show these results we prove also that if G is a connected, simply connected, semi-simple algebraic group defined over the field of rational numbers, then the G-representation variety of the fundamental group of a compact Riemann surface of genus n has ra…
Higher Jacobian ideals, contact equivalence and motivic zeta functions
本文研究高阶雅可比理想的性质，证明高阶Nash blowing-up代数在接触等价下不变，并得到动机近旁循环结果的类比，从而推进了函数奇点的研究。
In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $δ$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studi…
The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $μ(f)$ to $μ(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.
Our purpose is to investigate all defined Poincaré series associated with multi-index filtrations and value semigroups of curve singularities---not necessarily complex---with regard to the property of forgetting variables, i.e., by making variables of the series to be 1. Generalised Poincaré series of motivic nature will be also considered.
Lattice cohomology and the embedded topological type of plane curve singularities
本文提出解析格上同调这一约化曲线奇点新不变量，证明其对不可约平面曲线奇点是完备嵌入拓扑不变量，并在多分支情形下与整Seifert形式比较。
We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge-Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.
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