Homogeneous Singularities Research Articles

Extreme plurisubharmonic singularities

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in $\Rn$. Another class of extreme singularities is presented by means of a notion of relative type.

Weighted homogeneous singularities and rational homology disk smoothings

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weight.

Homogeneous singularity generated by exotic matter, with applications to collapsed black holes and wormholes

We analyze analytically and numerically the origin of the singularity in the course of the collapse of a wormhole with the exotic scalar field Psi with negative energy density, and with this field Psi together with the ordered magnetic field H. We do this under the simplifying assumptions of the spherical symmetry and that in the vicinity of the singularity the solution of the Einstein equations depends only on one coordinate (the homogeneous approximation). In the framework of these assumptions we found the principal difference between the case of the collapse of the ordinary scalar field Phi with the positive energy density together with an ordered magnetic field H and the collapse of the exotic scalar field Psi together with the magnetic field H. The later case is important for the possible astrophysical manifestation of the wormholes.

Singular ordinary differential equations homogeneous of degree $0$ near a codimension $2$ set

This work deals with an example of class of ordinary differential equations which are singular near a codimension 2 set, with an homogeneous singularity of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution and study the evolution of the Lebesgue measure of a transported set of initial data. The analysis is motivated by the Low Mach number asymptotics of compressible fluid models in the case of non isentropic flows, which involves such dynamical systems.

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV n ≃ ℂ2/ℤ n ,n ∈ {2,3,4,⋯} defined by the immersions of ℂ2 into ℂ n+1 X n : (z, w) → (z n ,z n−1 w, ⋯,zw n−1 ,w n )

The link of {f(x, y) + zn = 0} and Zariski's conjecture

We consider suspension hypersurface singularities of type g = f(x,y) + z n , where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

Generalized Tjurina numbers of an isolated complete intersection singularity

We define a series of non-negative integer valued invariants Open image in new window the generalized Tjurina numbers, of an analytic n-dimensional isolated complete intersection singularity. The Tjurina number τ equals τ(1). We show that all these numbers are bounded by the Milnor number μ by establishing explicit formulas for the differences μ−τ(p) involving the mixed Hodge numbers of the cohomology of the link and a new range of analytic invariants for (X,x). This generalizes earlier results of Looijenga and Steenbrink on the relation between the Milnor number and the Tjurina number. We show that singularities satisfying μ=τ(1)=⋯=τ(n−1) behave cohomologically like weighted homogeneous singularities, and we expect this to be a criterion for weighted homogeneity.

Algebraic classification of rational CR structures on topological 5-sphere with transversal holomorphic S-1-action in C-4

Let X be a compact connected CR manifold in ℂN. X is called a rational CR manifold if its geometric genus pg(X) is equal to zero. In this paper we classify all rational CR structures on a topological 5–sphere with transverse S1–action in ℂ4 up to algebraic equivalence. In fact the main theorem of this paper gives a classification of all 3–dimensional isolated rational weighted homogeneous singularities with links homeomorphic to topological 5–sphere. These are the simplest kind of singularities in some sense.

On the freeness of equisingular deformations of plane curve singularities

We consider surface singularities in C 3 arising as the total space of an equisingular deformation of an isolated curve singularity of the form f( x, y)+ zg( x, y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that the failure of this property “almost” characterizes the simple singularities. The criterion also yields distinct deformations of the same curve singularity, exactly one of which is free, showing that freeness is not a topological property.

HYPERSURFACE EXCEPTIONAL SINGULARITIES

This paper studies hypersurface exceptional singularities in [Formula: see text] defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.

Error expansions for multidimensional trapezoidal rules with Sidi transformations

In 1993, Sidi introduced a set of trigonometric transformations x = ψ(t) that improve the effectiveness of the one-dimensional trapezoidal quadrature rule for a finite interval. In this paper, we extend Sidi's approach to product multidimensional quadrature over [0,1] N . We establish the Euler–Maclaurin expansion for this rule, both in the case of a regular integrand function f(x) and in the cases when f(x) has homogeneous singularities confined to vertices.

Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous singularities

Nonexistence of negative weight derivation of moduli algebras of weighted homogeneous singularities

An error expansion for cubature with an integrand with homogeneous boundary singularities

A common strategy in the numerical integration over ann-dimensional hypercube or simplex, is to consider a regular subdivision of the integration domain intom n subdomains and to approximate the integral over each subdomain by means of a cubature formula. An asymptotic error expansion whenm ? ? is derived in case of an integrand with homogeneous boundary singularities. The error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains.

Durfee conjecture and coordinate free characterization of homogeneous singularities

This work is a natural continuation of our previous work [14]. The motivation of our work is to solve the Durfee conjecture. Let / : (C 3 , 0) —> (C, 0) be the germ of a complex analytic function with an isolated critical point at the origin. For e > 0 suitably small and δ yet smaller, the space V' = f~ι(δ)Γ\De (where D£ denotes the closed disk of radius e about 0) is a real oriented four-manifold with boundary whose diffeomorphism type depends only on / . It has been proved that V has the homotopy type of a wedge of two-spheres; the number μ of two-spheres is precisely d i m C { x 9 y 9 z } / ( f χ , f 9 f z ) . Let π: (AT, A) —> (V, 0) be a resolution of V = {(x,y, z) : f(x, y, z) = 0} with exceptional set A = π~ (0). The geometric genus p of the singularity V is the dimension of H(M, (?). Let χ(A) be the topological Euler characteristic of A , and K be the self-intersection number of the canonical divisor on M. Laufer's formula (cf. [5]) says that

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.

Topological invariance of weights for weighted homogeneous singularities

Topological invariance of weights for weighted homogeneous singularities

Quelques conséquences locales de la théorie de Hodge

A positivity result from Hodge theory permits us to give some poles of the distribution |f| 2z , for f an analytic function with isolated singularity. In the case of curves and of quasi homogeneous singularities the whole set of poles is given. We also prove that if the residue of a holomorphic form is square integrable on the special fiber, the integral on the special fibre is the limit of the integral on near by fibers.

The monodromy of weighted homogeneous singularities

Varieties defined by weighted homogeneous (or quasihomogeneous) polynomials are of particular interest because they admit actions of the multiplicative group of non-zero complex numbers [24] and because the topological properties of many general singularities may be computed from quasihomogeneous or semiquasihomogeneous normal forms [4]. In this paper we use hyperplane sections and relative monodromy to study such singularities, and we compute explicitly the local integral monodromy of a general class in any number of variables.