Counting isolated singularities in germs of applications ${\C^n,0\to \C^p,0}$, n<p

On finite determinacy of complete intersection singularities

We give an elementary combinatorial proof of the following fact: Every real or complex analytic complete intersection germ X is equisingular -- in the sense of the Hilbert-Samuel function -- with a germ of an algebraic set defined by sufficiently long truncations of the defining equations of X.

正常表面奇点上的严格萌芽

Strict germs on normal surface singularities

该研究证明了正常表面奇点间拓扑度为1的全纯萌芽可分解为修正与局部同构的组合，并探讨了自映射下的sandwiched奇点性质。

Hölder equivalence of complex analytic curve singularities

We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<α<1$ such that those germs are not $α$-Hölder homeomorphic. For germs of complex analytic plane curves with several irreducible components we prove that if any two of them are $α$-Hölder homeomorphic, for all $0<α<1$, then there is a correspondence between their branches preserving sequence of characteristic exponents and intersection multiplicity of pair of branches. In particular, we recovery the sequence of characteristic exponents…

Topological finite-determinacy of functions with non-isolated singularities

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal $I$, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let $A$ be the complement in the ideal $I$ of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion; then $A$ has infinite codimension in $I$ in a suitable sense. We also…

关于环面奇点的Lipschitz饱和

On the Lipschitz saturation of toric singularities

本文研究环面奇点芽的Lipschitz饱和，证明具有光滑正规化的环面奇点芽的Lipschitz饱和仍为环面奇点，并展示了如何从定义半群计算特定环面奇点族的Lipschitz饱和。

检测复解析芽中的快速消失环（以及检测内部度量锥形的芽）

Detecting fast vanishing loops in complex-analytic germs (and detecting germs that are inner metrically conical)

研究了纯维数n≥2的既约复解析芽中同调环的快速消失问题，利用判别式和覆盖数据给出了充要条件，并据此判定芽是否具有内部度量锥形（IMC）性质。

Classification and moduli spaces of dicritical singularities

In this paper we give complete analytic invariants for germs of holomorphic foliations in $(\mathbb{C}^2,0)$ that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize …

Reversible parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ and exceptional hyperbolic CR-singularities

The aim of this article is twofold: First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants. Their restriction to an irreductible component of the zero locus of the first integral reduces to the Birkhoff--Écalle--Voronin modulus of the 1-dimensional restricted parabolic germ.…

Topology of singular foliation germs in $\mathbb C^2$

The object of this survey is to give an overview on the topology of singularities of holomorphic foliation germs on $(\mathbb C^2,0)$.

Integrable deformations of local analytic fibrations with singularities

We study analytic integrable deformations of the germ of a holomorphic foliation given by $df=0$ at the origin $0 \in \mathbb C^n, n \geq 3$. We consider the case where $f$ is a germ of an irreducible and reduced holomorphic function. Our central hypotheses is that, {\em outside of a dimension $\leq n-3$ analytic subset $Y\subset X$, the analytic hypersurface $X_f : (f=0)$ has only normal crossings singularities}. We then prove that, as germs, such deformations also exhibit a holomorphic first integral, depending analytically on the parameter of the deformation. This applies to the study of in…

Nondegenerate germs of holomorphic foliations with prescribed holonomy

We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natu…

拟齐次非孤立奇点超曲面的Zariski重数猜想

Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities

本文针对2≤n≤4维拟齐次非孤立奇点超曲面，利用拟齐次正规形推导出由权重和次数决定的重数显式公式，证明重数在拓扑等价下不变，从而证实Zariski重数猜想。

Note on Milnor numbers of irreducible germs

Let $(\bf {V,0})\subset (\mathbb{C}^n,0)$ be a germ of a complex hypersurface and let $f: (\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ be a germ of a finite holomorphic mapping. If germs $(\bf {V,0})$ and ${\bf W}:=(F^{-1}(\bf{ V})),0)$ are irreducible and with isolated singularities, then $$μ(F^{-1}(\bf{ V}))\ge μ(\bf {V}),$$ where $μ$ denotes the Milnor number.

Pulling back singularities of codimension one objects

We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $g: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^n,0)$ is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal. Geom. 15 (2007), no. 2, 491-507], Lebl [arXiv:0812.2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1-2, 83-91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.

Explicit reconstruction of homogeneous isolated hypersurface singularities from their Milnor algebras

By the Mather-Yau theorem, a complex hypersurface germ $V$ with isolated singularity is completely determined by its moduli algebra $A(V)$. The proof of the theorem does not provide an explicit procedure for recovering $V$ from $A(V)$, and finding such a procedure is a long-standing open problem. In this paper, we present an explicit way for reconstructing $V$ from $A(V)$ up to biholomorphic equivalence under the assumption that the singularity of $V$ is homogeneous, in which case $A(V)$ coincides with the Milnor algebra of $V$.

Partial separatrices and local Brunella's alternative

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in $({\mathbb C}^3,0)$ without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argu…

Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valuations

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…

孤立奇点的等变形及其应用

Equivariant deformations of isolated singularities and applications

本文发展了复李群作用下复空间芽形变理论的等变版本，证明了约化情形下等变半泛形变的存在性，并通过反例说明非约化情形下一般不存在，推广了Grauert-Donin和Ferrer-Puerta-Slodowy的定理，并应用于对形变、刚性和Milnor纤维作用。

Stratification of Unfoldings of Corank 1 Singularities

In the study of equisingularity of families of mappings Gaffney introduced the crucial notion of excellent unfoldings. This definition essentially says that the family can be stratified so that there are no strata of dimension 1 other than the parameter axis for the family. Consider a family of corank 1 multi-germs with source dimension less than target. In this paper it is shown how image Milnor numbers can ensure some of the conditions involved in being excellent. The methods used can also be successfully applied to cases where the double point set is a curve. In order to prove the results t…

奇点的增广：$μ/τ$-型猜想与简单性

Augmentation of singularities: $μ/τ$-type conjectures and simplicity

本文研究了映射芽增广的 $\mu/\tau$-型猜想，给出了其余维数上界并证明了相关不等式，同时探讨了增广的刻画、开折独立性及简单性条件。

Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities

The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically staightened. The second part is devoted to the study real analytic $(n+r)$-submanifolds of $(\Bbb C^n,0)$ having a CR-singularity at the origin ($r$ is a nonnegative integer). We consider deformations of quadrics and we define genera…

Isolated hypersurface singularities and polynomial realizations of affine quadrics

Let $V$, $\tilde V$ be hypersurface germs in $\CC^m$, each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for $V$, $\tilde V$ reduces to the linear equivalence problem for certain polynomials $P$, $\tilde P$ arising from the moduli algebras of $V$, $\tilde V$. The polynomials $P$, $\tilde P$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for $V$, $\tilde V$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.

Mixed tête-à-tête twists as monodromies associated with holomorphic function germs

Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced functi…