Françoise Michel
If (X,0) is a complex surface germ with a non-isolated singular locus we describe its singular link L of (X,0) and we show that the topology of L determines the topology of the normalization.
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We construct all codimension 1 multi-germs of maps (k^n,T)-->(k^p,0) with n > p-2, (n,p) nice dimensions, k = R or C, by augmentation and concetenation operations, starting from mon-germs (|T|=1). As an application, we prove general results for multi-germs of corank <2: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n=p-1 every one has image Milnor number equal to 1 (the comparable result for n>p-1 being already known).
In a previous paper the authors elaborated notions and technique which could be applied to compute such invariants of polynomials as Euler characteristics of fibres and zeta-functions of monodromy transformations associated with a polynomial. Some crucial basic properties of the notions related to the topology of meromorphic germs were not discussed there. This has produced some lack of understanding of the general constructions. The aim of this note is to partially fill in this gap. At the same time we describe connections with some previous results and generalizations of them.
Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of $G$-sets with an additional structure. We discuss to which extend the $G$-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré s…
To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated …
The Topology of Surface Singularities
本文研究复表面奇点的拓扑结构，通过分析表面germ的链接来描述其拓扑特性。
We consider a corank $1$, finitely determined, quasi-homogeneous map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$, denoted by $γ_f$, in terms of the weights and degrees of $f$. As a consequence, a necessary condition for a corank $1$ finitely determined map germ $g:(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $γ_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfo…
Logarithmic Jacobian ideals, quasi-ordinary hypersurfaces and equisingularity
针对不可约超曲面芽的拟平凡性，研究其雅可比理想扩张为单项式理想的性质，通过与关联单项式簇的对数雅可比理想比较进行描述并给出应用。
Inner Lipschitz Geometry of Complex Surface Germs with Non-Isolated Singularities: A Complete Classification
本文通过归一化映射坐标函数的数值不变量和适当好解消的组合学，给出了非孤立奇点复曲面芽内部Lipschitz几何的完全不变量，将Birbrair-Neumann-Pichon的结果推广到非孤立情形。
On the topology of complex map-germs and a general Lê-Greuel formula
本文利用分层 Morse 理论，为奇异复映射芽建立了比较 Milnor 纤维 Euler 特征的通用 Lê-Greuel 公式，统一了 ICIS、曲线、曲面、行列式奇点及像 Milnor 数等已知公式。
In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.
Lipschitz geometry of complex surface germs via inner rates of primary ideals
本文通过为准素理想关联连续函数，证明了该函数由复曲面芽的外部Lipschitz几何决定，并构造了正规化同构但Lipschitz类型不同的孤立奇点无穷族。
In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. We describe the link $L_F$ as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ $f \bar{g}$ and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ $F$ gives rise to a Milnor fibration $\frac{F}{|F|…
We associate to any irreducible germ S of complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the normalized characteristic exponents. These exponents generalize the generic Newton-Puiseux exponents of plane curves. Incidentally, we give a toric description of the normalization morphism of the germ S.
An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.
Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres of a real weighted-homogeneous polynomial and a real version of the L{ê}-Iomdine formula. We have also included some results of the same flavor on the local topology of locally closed definable sets.
In 1982, Tamaki Yano proposed a conjecture predicting how is the set of $b$-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In 1986, Pi.~Cassou-Noguès proved the conjecture for the one Puiseux pair case. In a previous work the authors proved the conjecture for two Puiseux pairs germs whose complex algebraic monodromy has distinct eigenvalues. A natural problem induced by Yano's conjecture is, for a generic equisingular deformation of an isolated plane curve singularity germ to study how the set of $b$-exponents depends on the topology of …
We study the boundary $L_t$ of the Milnor fiber for the reduced holomorphic germs $f:(\Bbb C^3,0) \rightarrow (\Bbb C,0)$ having a non-isolated singularity at $0$. We prove that $L_t$ is a graph manifold by using a new technique of carrousel depending on one parameter varying on a circle. Our results enable one to compare the topology of $L_t$ and of the link of the normalization of $f^{-1}(0)$.
In this work we study some algebraic and topological properties of the ring ${\mathcal O}(X^ν)$ of global analytic functions of the normalization $(X^ν,{\mathcal O}_{X^ν})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize ${\mathcal O}(X^ν)$ in terms of the (topological) completion of the integral closure $\overline{{\mathcal O}(X)}^ν$ of the ring ${\mathcal O}(X)$ of global holomorphic functions on $X$ (inside its total ring of fractions) with respect to the usual Fréchet topology of $\overline{{\mathcal O}(X)}^ν$. This shows …
We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topol…
In this article, we study the topology of real analytic germs $F \colon (\C^3,0) \to (\C,0)$ given by $F(x,y,z)=\overline{xy}(x^p+y^q)+z^r$ with $p,q,r \in \N$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ gives rise to a Milnor fibration $\frac{F}{\mid F \mid} \colon \Sp^5\setminus L_F \to \Sp^1$. We describe the link $L_F$ as a Seifert manifold and we show that in many cases the open-book decomposition of $\Sp^5$ given by the Milnor fibration of $F$ cannot come from the Milnor fibration of a complex singularity in $\C^3$.
Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together wit…
The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold cyclic covering of a surface germ, branched along a curve given by the germ of an analytic function f. We present non-trivial examples in order to show that from the embedded resolution graph G of f in general it is not possible to recover the resolution graph of the cyclic covering. The main results are the construction of a ``universal covering graph'' from t…
Topological and bilipschitz types of complex surface singularities and their links
本文证明正规复曲面芽的内双Lipschitz同胚蕴含相同定向拓扑型与最小配管图，并给出锥形结构定理的逆命题，为此研究了Hirzebruch-Jung奇点与尖点奇点的拓扑及内Lipschitz几何。
We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine tor…
We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface singularities of [10]. Our thin zone catches exactly the homology of the family of the links collapsing faster than linearly. Simultaneously we introduce a class of rigid homeomorphisms more general than bi-Lipschitz ones, which map the topological thin zone onto the topological thin zone of its image. As a consequence of this point of view for the class of singula…
We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in ${\bf C}^3$. We study irreducible (i.e. $gcd (m,k,l) = 1$) non-isolated (i.e. $1 \leq k < l$) Hirzebruch hypersurface singularities in ${\bf C}^3$ given by the equation $z^m - x^ky^l = 0$. We show that the boundary $L$ of the Milnor fiber is always a Seifert manifold and we give an explicit description of the Seifert structure. From it, we deduce that : 1) $L$ is never diffeomorphic to the boundary of the normalization. 2) $L$ is a lens space iff $m…
We examine complements (inside products of a smooth projective complex curve of arbitrary genus) of unions of diagonals indexed by the edges of an arbitrary simple graph. We use Gysin models associated to these quasi-projective manifolds to compute pairs of analytic germs at the origin, both for rank 1 and 2 representation varieties of their fundamental groups, and for degree 1 topological Green--Lazarsfeld loci. As a corollary, we describe all regular surjections with connected generic fiber, defined on the above complements onto smooth complex curves of negative Euler characteristic. We show…
Images of maps from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^n,0)$
该研究针对从二维复空间原点到n维复空间原点的有限映射芽的像，借助链环拓扑证明正规像必为商奇点并刻画其拓扑类型，同时探讨非正规像的情形，为复奇点拓扑研究提供新结论。
Let (C,0) be a reduced curve germ in a normal surface singularity (X,0). The main goal is to recover the delta invariant of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if (X,0) is a quotient singularity, then the delta invariant of (C,0) only admits the values r-1 or r, where r is the number or irreducible components of (C,0). (r-1 realizes the extremal lower bound, valid only for `ordinary r-tuples'.)
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