Stable Germs Research Articles

The extra-nice dimensions

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in C^{infty }(Ntimes [0,1],P) is dense if and only if the pair of dimensions (dim N, dim P) is in the extra-nice dimensions. This result is parallel to Mather’s characterization of the nice dimensions as the pairs (n, p) for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have {mathscr {A}}_e-codimension 1 hyperplane sections. They are also related to the simplicity of {mathscr {A}}_e-codimension 2 germs. We give a sufficient condition for any {mathscr {A}}_e-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.

Local classification of stable geometric solutions of systems of quasilinear first-order PDE

Systems of quasilinear first order PDE are studied in the framework of contact manifold. All of the local stable geometric solutions of such systems are classified by using versal deformation and the classification of stable map germs of type ∑ 1 in singularity theory.

$$A$$ and the vanishing topology of discriminants

Suppose thatf: ℂn, 0→ℂp, 0 is finitely\(A\)-determined withn≧p. We define a “Milnor fiber” for the discriminant off; it is the discriminant of a “stabilization” off. We prove that this “discriminant Milnor fiber” has the homotopy type of a wedge of spheres of dimensionp−1, whose number we denote byµΔ(f). One of the main theorems of the paper is a “μ=τ” type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then\(\mu _\Delta (f) \geqq A_e \)-codium,with equality iff is weighted homogeneous. Outside the nice dimensions we obtain analogous formulae with correction terms measuring the presence of unstable but topologically stable germs in the stabilization. These results are further extended to nonlinear sections of free divisors.

Stable mappings of discriminant varieties

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

Typical (n-1)-stable germs of Pfaff equations in Rn

Typical (n-1)-stable germs of Pfaff equations in Rn

Topological properties of real simple germs, curves, and the nice dimensions n > p

Infinitesimally stable germs play an important role both as germs from which global C∞-stable mappings are constructed and as germs representing versal unfoldings of (C∞ or holomorphic) germs. Because of the presence of moduli, the C∞ (or analytic) classification of these germs is insufficient and the topological classification of these germs must be understood as well. Here we consider the classification of such germs in the region where no moduli occur. This region is important for several reasons. Most importantly, it contains the infinitesimally stable germs occurring in the nice dimensions. These are the dimensions in which globally infinitesimally stable mappings are dense among the proper C∞-mappings. In (4), it was proved that the topological and C∞-classifications agree for infinitesimally stable germs f: n, 0 → p, 0 in the nice dimensions, n ≤ p. This was then used in (5) to characterize those topologically stable germs which are C∞-stable.

Flache T1-Stabilit�t in der Strati-fizierung verseller Entfaltungen

Let 0 be the local ring of a simple singularity defined over the complex numbers and τ the dimension of its versal deformation space. Than it is well known that any nearby singularity in this space is also simple and has smaller unfolding dimension in the hierarchy of simple singularities. In particular this implies that the τ=max-stratum consists just of one point namely the given singularity. We want to generalize this concept as we are interested in families of varieties with formal unchanged singularities. For this we introduce in quite generality the notion of flat T1-stabi1ity which may be checked for any k- algebra 0 where k is for simplicity an algebraically closed field of a priori arbitrary characteristics. We call 0 formal flat T1 stable or for short flat T1-stable if the following is true: if R is any deformation of 0 over an Artin local finite k-algebra A and if T1(R/A,R) is A-flat than R is isomorphic to the trivial deformation\(0\mathop \otimes \limits_K A\). T1(R/A,R) is the first cotangent module of R over A with values in R. Obviously the simple singularities Ak, Dk, E6, E7, E8 fulfill this criterion over C but we look also at fibres of arbitrary stable map germs, generic singularities of algebraic varieties where we have to modify this notion in order to deal with wild ramification and to quasihomo-genous hypersurface singularities where it functorializes because in this case T1 commutes with arbitrary base change. The notion of flat T1-stable singularities is closely related to questions of existence of equisingular families and is used in[12] and [5], [6] to stratify certain Hilbert schemes.

The Structure of the Thom-Boardman Singularities of Stable Germs with Type Σ2, 0

The Structure of the Thom-Boardman Singularities of Stable Germs with Type Σ^{2, 0}

On the order of determination of a finitely determined germ

A map germ f is said to be determined of order k if f is smoothly equivalent to its Taylor polynomial of order k, and any g with the same Taylor polynomial of order k is smoothly equivalent to f A map germ is finitely determined if it is determined of some order k. Given a finitely determined germ f a question immediately arises as to the order of the Taylor polynomial which determines f. The only previous estimate for finitely determined non stable germs is due to Mather and the estimate is astronomical [2]. This paper provides a low order estimate of the order of deter- mination in terms of the power of the maximal ideal contained by tf (O(n)) + o)f (O(p)) and p the dimension of the target space. (1.4) Theorem. Suppose f is in Eo(n,p) and tf(O(n))+mf(O(p)) contains mkO(f), k > O; then f is k(p + 1) determined. (Recall that iff is finitely determined tf(O(n)) + ~of (O(p)) must contain m k O(f) for some k.) Another variant of the generalized Malgrange preparation theorem is used as a tool in the proof. (2.3) Theorem. (Generalized Preparation Theorem). Let f: (IR", 0)-* (~P, 0), g: (~,, 0)--~ (~P', 0) be Coo map germs. Let A be a finitely generated (f, g)* Cp+p, module. If A/g* (rap,) A is finite dimensional as an lR vector space, then A is finitely generated as a g*(Cp,) module. The notation used in this paper is essentially the same as John Mather's papers on stability of C ~ mappings: d is the group of coordinate changes in source and target. Eo(n, p) stands for orign preserving map germs from either ~" to IR p or from I~" to IE p which are either Coo or analytic in the first case or analytic in the second case. C, stands for Coo germs f: (IR", 0)~ IR". Iff is in Eo(n, p), tf(O(n)) is the set of p tuples of germs obtained by applying the differential of f to the set of n tuples of germs in E, ; o)f(O(p)) is the set of p

The nonexistence of globally stable forms

It is shown that on a closed manifold there are no globally stable differential forms. A number of people have worked on the problem of the stability of differential forms. Martinet [1] has inspected the singularities and stability of germs of p-forms. His definition for stability is the following: a germ of a pform X is stable if for every nearby germ o' there is a germ of a diffeomorphism f such that f * &' = w. In this paper Martinet computes some examples of stable germs. The stability of globally defined closed differential forms where the nearby forms o' are allowed to vary only within the cohomology class of X have been studied by Moser [2], Chatelet and Rosenberg [3], and others. A very tempting idea-given Martinet's sucess-is to try to find globally stable forms on a compact manifold M using the following: DEFINITION 1. A p-form X on a manifold X is stable if there is a neighborhood Q of X in the C? topology on p-forms such that if C' is in Q, then there is a diffeomorphism f: X -* X such that f* C' = w. Unfortunately, this definition of stability for p-forms does have problems, for a little thought shows that there are obstructions to the existence of globally defined stable forms. In fact we show: THEOREM 2. Using this definition of global stability, there do not exist globally stable forms on compact manifolds. The authors wish to thank Mike Shub and Victor Guillemin for helpful conversations. Our arguments fall into two classes: first we show that in a large number of cases of p-forms on n-manifolds even local stability for germs in the sense of Martinet is not possible; second, in the remaining cases there are global invariants which obstruct the existence of stable forms. LEMMA 3. On an n-manifold where n > 10, there exist no locally or globally stable p-forms where 3 < p < n 3. The same is true for 4or 5-forms on 9manifolds. First, some notation. Let Diff(X) denote the group of C? diffeomorphisms Received by the editors December 30, 1974 and, in revised form, October 27, 1975. AMS (MOS) subject classifications (1970). Primary 58A10; Secondary 57D45. I Research partially supported by NSF Contract no. GP 43524 and by a Faculty Research Grant of CUNY. (D Amencan Mathematical Society 19/6

Plane stable germs of C?-maps and their linear approximations

Plane stable germs of C?-maps and their linear approximations

Stability of C∞ mappings, IV: Classification of stable germs by R-algebrasmappings, IV: Classification of stable germs by R-algebras

Stability of C∞ mappings, IV: Classification of stable germs by R-algebrasmappings, IV: Classification of stable germs by R-algebras