10 pages, corrects typos and a small mistake in the proof of theorem 1.1. Several proofs are expanded to improve readability. Material on the Euler obstruction added
The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the simplest D(q,p) singularity is the Whitney umbrella. These are the basic examples, because they correspond to stable germs of functions in the study of germs of functions with non-isolated singularities. Given a germ of a function which defines a non-isolated hypersurface singularity at the origin, which in the appropriate sense, has finite codimension in the set of such germs, the singularity type of such germs away from the origin is A(d) or D(q,p).
In this note we calculate the homotopy type of the Milnor fiber of germs of type D(q,p), as well as their Lê numbers. The calculation of the Lê numbers involves the use of an incidence variety which may be useful for studying germs of finite codimension. The calculation shows that the set of symmetric matrices of kernel rank greater than or equal to 1 is an example of a hypersurface singularity with a Whitney stratification (given by the rank of the matrices) in which only one singular stratum gives a component of top dimension of the singular set of the conormal.
In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities.
Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ given by the multiplication of f by the conjugate of g has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs given by the multiplication…
In this paper we give a formula for counting the number of isolated stable singularities of a stable perturbation of corank 1 germs $f:\C^n,0\to \C^p,0$ with $n<p$ that appear in the image $f(\C^n).$ We also define a set of ${\cal A}$-invariants and show that their finiteness is a necessary and sufficient condition for the ${\cal A}$-finiteness of the germ $f.$
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…
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…
A germ of normal complex analytical surface is called a Hirzebruch-Jung singularity if it is analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly isomorphic. We extend this result to higher-dimensional Hirzebruch-Jung singularities, which we define to be the germs analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric variety determined by a lattice and a simplicial cone of maximal dimension. We deduce a normalizati…
In this paper we study the problem of analytic extension of germs of holonomy of algebraic foliations. More precisely we prove that for a Riccati foliation associated to a branched projective structure over a finite type surface which is non-elementary and parabolic, all the germs of holonomy between a fiber and a holomorphic section of the bundle are led to singularities by almost every developed geodesic ray. We study in detail the distribution of these singularities and prove in particular that they are included and dense in the limit set and uncountable, giving another negative answer to a…
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.
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.
In this work, we consider a quasi-homogeneous, corank $1$, finitely determined map germ $f$ from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$. We consider the invariants $m(f(D(f))$ and $J$, where $m(f(D(f))$ denotes the multiplicity of the image of the double point curve $D(f)$ of $f$ and $J$ denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of $f(\mathbb{C}^2)$. We present formulas to calculate $m(f(D(f))$ and $J$ in terms of the weights and degrees of $f$.
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…
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 …
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…
Mather and Yau showed that an isolated complex hypersurface singularity is completely determined by its moduli algebra. It is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and, hence, associate invariants to the singularities themselves.
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.…
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…
We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way of extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.
It was recently proved that for finitely determined germs $ Φ: ( \mathbb{C}^2, 0) \to ( \mathbb{C}^3, 0) $ the number $C(Φ)$ of Whitney umbrella points and the number $T(Φ)$ of triple values of a stable deformation are topological invariants. The proof uses the fact that the combination $C(Φ)-3T(Φ)$ is topological since it equals the linking invariant of the associated immersion $S^3 \looparrowright S^5$ introduced by Ekholm and Szűcs. We provide a new, direct proof for this equality. We also clarify the relation between various definitions of the latter invariant.
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…
mu-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo {id,-id}. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these modu…
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…
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…
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 $V$.
Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff$(\bf C,0)$. We also show that two nonlinearizable germs are conjugate if and only if their rotation numbers are equal and a hedgehog of one can be mapped conformall…
We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain polynomials associated to such algebras, called nil-polynomials, and we compare them with two other classes of polynomials that have also been used to produce invariants.
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.
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…
