Bi-Lipschitz geometry of complex surface singularities

Lipschitz Normal Embeddings and Determinantal Singularities

The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove that the model determinantal singularity, that is the space of $m\times n$ matrices of rank less than a given number, is Lipschitz normally embedded. We will also discuss some of the difficulties extending this result to the case of general determinantal singularities.

Ambient Lipschitz equivalence of real surface singularities

We present a series of examples of pairs of singular semialgebraic surfaces (real semialgebraic sets of dimension two) in ${\mathbb R}^3$ and ${\mathbb R}^4$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset {\mathbb R}^4$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to $S$, but pairwise ambient Lipschitz non-equivalent.

The Bi-Lipschitz Equisingularity of Essentially Isolated Determinantal Singularities

The bi-Lipschitz geometry is one of the main subjects in the modern approach of Singularity Theory. However, it rises from works of important mathematicians of the last century, especially Zariski. In this work we investigate the Bi-Lipschitz equisingularity of families of Essentially Isolated Determinantal Singularities inspired by the approach of Mostowski and Gaffney.

Lipschitz normal embedding among superisolated singularities

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if…

An introduction to Lipschitz geometry of complex singularities

The aim of this paper to introduce the reader to a recent point of view on the Lipschitz classifications of complex singularities. It presents the complete classification of Lipschitz geometry of complex plane curves singularities and in particular, it introduces the so-called bubble trick and bubble trick with jumps which are key tools to study Lipschitz geometry of germs. It describes also the thick-thin decomposition of a normal complex surface singularity and built two geometric decompositions of a normal surface germ into standard pieces which are invariant by respectively inner and outer…

Minimal surface singularities are Lipschitz normally embedded

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.

代数几何与交换代数中任意维数的托里奇奇点的Lipschitz饱和

Lipschitz saturation of toric singularities in any dimension

本文研究了任意维数复解析托里奇奇点的Lipschitz饱和的半群，给出了单调多项式属于该饱和的充要条件，并推导出有限算法。同时指出在二维以上，Campillo的预饱和与Lipschitz饱和存在差异。

A characterization of Lipschitz normally embedded surface singularities

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), i.e., to have bilipschitz equivalent outer and inner metrics. In a partner paper [15] we apply it to prove that rational surface singularities are LNE if and only if they are mini…

Lipschitz单模函数芽的分类

Classification of Lipschitz unimodal function germs

本文为孤立奇点引入Lipschitz模概念，完整分类了余秩2且4-jet非零的Lipschitz单模奇点，并证明6-jet消失时模至少为2，给出上界。

关于Lipschitz正规嵌入奇点

On Lipschitz Normally Embedded singularities

本文综述了Lipschitz正规嵌入奇点的一般理论，并重点介绍了复曲面情形下的判据、例子和性质，最后提出若干开放问题。

非孤立奇点复曲面芽的内部Lipschitz几何：完全分类

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。

mu-constancy does not imply constant bi-Lipschitz type

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\ne 0$.

Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.

Lipschitz geometry does not determine embedded topological type

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs $(X_1,0)$ and $(X_2,0)$ in $(\mathbb C^3,0)$ having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones…

Lipschitz geometry of surface germs in $\mathbb{R}^4$: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that: the link at the origin of $X_{K}$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. We s…

Lipschitz Geometry of Real Semialgebraic Surfaces

We present here basic results in Lipschitz Geometry of semialgebraic surface germs. Although bi-Lipschitz classification problem of surface germs with respect to the inner metric was solved long ago, classification with respect to the outer metric remains an open problem. We review recent results related to the outer and ambient bi-Lipschitz classification of surface germs. In particular, we explain why the outer Lipschitz classification is much harder than the inner classification, and why the ambient Lipschitz Geometry of surface germs is very different from their outer Lipschitz Geometry. I…

Choking horns in Lipschitz Geometry of Complex Algebraic Varieties

We study the Lipschitz Geometry of Complex Algebraic Singularities. For this purpose we introduce the notion of choking horns. A Choking horn is a family of cycles on the family of the sections of an algebraic variety by very small spheres centered at a singular point, such that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness as we can see with some classical isolated hypersurfaces singularities which we prove are not metrically conic. We also show that there exist infinitely countably many singular varieties, which are lo…

Lipschitz geometry of complex curves

We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, but in an analytic category.

Lipschitz geometry of complex plane algebraic curves

We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.

Lipschitz Stratification of Complex Hypersurfaces in Codimension 2

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated, singularities in $\mathbb{C}^3$. We show that a natural stratification of such a family given by the singular set and the generic family of polar curves provides a Lipschitz stratification in the sense of Mostowski. In particular, such families are bi-Lipschitz trivial by trivializations obtained by integrating Lipschitz vector fields.

Local Lipschitz geometry of weighted homogeneous surfaces

We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.

正规嵌入曲面芽的环境利普希茨几何

Ambient Lipschitz geometry of normally embedded surface germs

本文研究半代数曲面的环境利普希茨几何，证明ℝ³中具孤立奇点的利普希茨正规嵌入曲面芽的环境双利普希茨等价性等价于外部双利普希茨等价性与环境拓扑等价性的结合。

Global bi-Lipschitz classification of semi-algebraic surfaces

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature.

实与复积分闭包、Lipschitz 等奇异性及其在方阵上的应用

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

本文改进矩阵芽 Lipschitz 平凡性的已有结果，将积分闭包结论推广到实背景，并应用于 Bruce 和 Tari 分类的方阵奇异性研究。

具有锥奇异的子流形的全局 Lipschitz 几何及其在代数集中的应用

Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic sets

本文证明了黎曼流形中连通的锥奇异子流形是 Lipschitz 正常嵌入的，并利用单点紧化下的闭包等价条件，推导出一般仿射实与复代数集的连通分支在无穷远处呈锥形且满足该嵌入性质。

Lipschitz geometry of pairs of normally embedded Hölder triangles

We consider a special case of the outer bi-Lipschitz classification of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure) surface germs, obtained as a union of two normally embedded Hölder triangles. We define a combinatorial invariant of an equivalence class of such surface germs, called $στ$-pizza, and conjecture that, in this special case, it is a complete combinatorial invariant of outer bi-Lipschitz equivalence.

Bi-Lipschitz geometry of contact orbits in the boundary of the nice dimensions

Mather proved that the smooth stability of smooth maps between manifolds is a generic condition if and only if the pair of dimensions of the manifolds are 'nice dimensions' while topologically stability is a generic condition in any pair of dimensions. And, by a result of du Plessis and Wall $C^1$-stability is also a generic condition precisely in the nice dimensions. We address the question of bi-Lipschitz stability in this article. We prove that the Thom-Mather stratification is bi-Lipschitz contact invariant in the boundary of the nice dimensions. This is done in two steps: first we explici…

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

It was already known that a p-adic, locally Lipschitz continuous semi-algebraic function is piecewise Lipschitz continuous, where the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1. We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative…

具有非孤立奇点的三次三重流形的等变几何

Equivariant geometry of cubic threefolds with non-isolated singularities

本文研究有限群在具有非孤立奇点的三次三重流形上作用的可线性化性问题，属于等变几何范畴。

The geometry of continued fractions and the topology of surface singularities

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers l >1 and l/(l…

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps app…

拟$F$-奇点与双有理几何中的奇点

Quasi-$F$-singularities and singularities in birational geometry

综述拟$F$-奇点理论，重点阐述其与双有理几何中奇点的联系。

Inner Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn.

Geometry of non-degenerate locally tame non-isolated singularities

We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.

Singularities of theta divisors in algebraic geometry

The singularities of theta divisors have played an important role in the study of algebraic varieties. This paper surveys some of the recent progress in this subject, using as motivation some well known results, especially those for Jacobians.

On Lipschitz Normally Embedded complex surface germs

We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.

Exceptional rays and bilipschitz geometry of real surface singularities

It is known that ambient bilipschitz equivalence preserves tangent cones. This paper explores the behavior of the Nash cone and, in particular, exceptional rays under ambient bilipschitz equivalence for real surfaces in $R^3$ with isolated singularity.

Lipschitz geometry and combinatorics of abnormal surface germs

We study outer Lipschitz geometry of real semialgebraic or, more general, definable in a polynomially bounded o-minimal structure over the reals, surface germs. In particular, any definable Hölder triangle is either Lipschitz normally embedded or contains some "abnormal" arcs. We show that abnormal arcs constitute finitely many "abnormal zones" in the space of all arcs, and investigate geometric and combinatorial properties of abnormal surface germs. We establish a strong relation between geometry and combinatorics of abnormal Hölder triangles.

On the decomposition of a 2D-complex germ with non-isolated singularities

The decomposition of a two dimensional complex germ with non-isolated singularity into semi-algebraic sets is given. This decomposition consists of four classes: Riemannian cones defined over a Seifert fibered manifold, a topological cone over thickened tori endowed with Cheeger-Nagase metric, a topological cone over mapping torus endowed with Hsiang-Pati metric and a topological cone over the tubular neighbourhoods of the link's singularities. In this decomposition there exist semi-algebraic sets that are metrically conical over the manifolds constituting the link. The germ is reconstituted u…

Bilipschitz几何：切锥为平面的实曲面奇点

Bilipschitz geometry of real surface singularities whose tangent cone is a plane

本文研究实曲面奇点在双Lipschitz等价下Nash锥与例外射线的行为，针对切锥为平面的孤立奇点情形。

具有孤立奇点的超曲面的几何

Geometry of Hypersurfaces with Isolated Singularities

本文研究具有孤立奇点的超曲面中直线的Fano簇，证明过奇点的直线簇的不可约性，并研究多奇点三次超曲面的直线Fano簇。

Birational geometry of del Pezzo fibrations with terminal quotient singularities

Del Pezzo fibrations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo fibrations is a well studied question but smooth fibrations are not dense in moduli. Little is known about the rationality of the singular models. We prove birational rigidity, hence non-rationality, of del Pezzo fibrations with simple non-Gorenstein singularities satisfying the famous $K^2$-condition. We then apply this result to study embeddings of $\operatorname{PSL}_2(7)$ into the Cremona group.

On discriminants, Tjurina modifications and the geometry of determinantal singularities

We describe a method for computing discriminants for a large class of families of isolated determinantal singularities -- more precisely, for subfamilies of ${\mathcal G}$-versal families. The approach intrinsically provides a decomposition of the discriminant into two parts and allows the computation of the determinantal and the non-determinantal loci of the family without extra effort; only the latter manifests itself in the Tjurina transform. This knowledge is then applied to the case of Cohen-Macaulay codimension 2 singularities putting several known, but previously unexplained observation…

奇点解消与判别式几何中亟待澄清的一些想法

Some ideas in need of clarification in resolution of singularities and the geometry of discriminants

本文提出了代数几何两个领域的问题：正特征下与非Abhyankar赋值相关的奇点解消，以及Thom意义下实极小通用展开判别式几何与配边上Morse函数运动的关系。

On Lipschitz Geometry at infinity of complex analytic sets

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which i…

The singularities and birational geometry of the universal compactified Jacobian

In this paper we establish that the singularities of the universal compactified Jacobian are canonical if the genus is at least four. As a corollary we determine the Kodaira dimension and the Iitaka fibration of the universal compactified Jacobian for every degree and genus. We also determine the birational automorphism group for every degree if the genus is at least twelve. This extends work of G. Farkas and A. Verra, as well as that of G. Bini, C. Fontanari and the third author.

Moduli of $G$-covers of curves: geometry and singularities

In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ of curves with a $G$-cover for any finite group $G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space $\overline{\mathcal M}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, th…

Singularities and Poisson geometry of certain representation spaces

Playing off against each other the real and complex structures, we elucidate the local structure of certain representation spaces in the world of Poisson geometry. Particular cases of these spaces arise as moduli spaces of semistable holomorphic vector bundles on Riemann surfaces.

从平面曲线奇点的计数几何角度看对数凹性

Log-concavity from enumerative geometry of planar curve singularities

本文提出关于平面曲线奇点计数几何中BPS不变量的对数凹性猜想，并将其扩展到Legendrian链环的规则多项式和特征簇的E多项式，证明了不可约加权齐次奇点（扭线结）和ADE奇点的猜想，以及规则多项式的乘法性质。

复曲面奇点及其链环的拓扑型与双Lipschitz型

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。

Regular vectors and bi-Lipschitz trivial stratifications in o-minimal structures

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which is locally definably bi-Lipschitz trivial along the strata. This result is obtained as a byproduct of two foregoing results of the author. The first one asserts that, given a family definable in an o-minimal structure, there is a regular vector, up to a definable family of bi-Lipschitz homeomorphisms. The second one is a bi-Lipschitz version of the famous H…

局部与全局 Lipschitz 几何

Local vs. global Lipschitz geometry

本文研究 o-minimal 结构中可定义集合链的内蕴距离等价性，并据此建立奇点局部与全局 Lipschitz 几何之间的联系，刻画外部 lipeomorphism、球极投影修改与反演之间的等价关系。

tt* geometry, Frobenius manifolds, their connections, and the construction for singularities

The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt* geometry if the singularity is quasihomogeneous. tt* geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6-8) Frobenius manifolds and tt* geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed …

关于次解析几何

On subanalytic geometry

本文是一篇次解析几何综述，梳理了全局次解析集的定义、基础工具、利普希茨几何进展及几何积分理论相关成果。