TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES

2021 ◽  
pp. 1-15
Author(s):  
NICOLAS DUTERTRE ◽  
JUAN ANTONIO MOYA PÉREZ
Keyword(s):  
Analytic Function ◽  
Euler Characteristic ◽  
Topological Degree ◽  
Isolated Singularities ◽  
Function Germ ◽  
Milnor Fiber ◽  
Real Singularities ◽  
Parameter Deformation

Abstract Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

scholarly journals Invariants of topological relative right equivalences

2013 ◽  
Vol 155 (2) ◽  
pp. 307-315 ◽  
Author(s):  
IMRAN AHMED ◽  
MARIA APARECIDA SOARES RUAS ◽  
JOÃO NIVALDO TOMAZELLA
Keyword(s):  
Analytic Function ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Function Germ ◽  
Analytic Variety ◽  
Relative Version ◽  
Topological Invariance

AbstractLet (V,0) be the germ of an analytic variety in $\mathbb{C}^n$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

scholarly journals Mixed Bruce–Roberts numbers

2020 ◽  
Vol 63 (2) ◽  
pp. 456-474 ◽  
Author(s):  
Carles Bivià-Ausina ◽  
Maria Aparecida Soares Ruas
Keyword(s):  
Analytic Function ◽  
Function Germ ◽  
Analytic Variety ◽  
Fundamental Properties ◽  
Complex Analytic Variety

AbstractWe extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂn and a finitely ${\mathcal R}(X)$-determined analytic function germ f : (ℂn, 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

scholarly journals INTERSECTION COHOMOLOGY, MONODROMY AND THE MILNOR FIBER

2009 ◽  
Vol 20 (04) ◽  
pp. 491-507 ◽  
Author(s):  
DAVID B. MASSEY
Keyword(s):  
Analytic Function ◽  
Simple Relation ◽  
Intersection Cohomology ◽  
Critical Locus ◽  
One Dimensional ◽  
Milnor Fiber ◽  
Singular Sets ◽  
Vanishing Cycle ◽  
Thom Isomorphism ◽  
Complex Analytic Space

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

On the Milnor Fiber of a Quasi-ordinary Surface Singularity

2002 ◽  
Vol 54 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Chunsheng Ban ◽  
Lee J. McEwan ◽  
András Némethi
Keyword(s):  
Explicit Formula ◽  
Euler Characteristic ◽  
Homotopy Type ◽  
Topological Type ◽  
Surface Singularity ◽  
Hypersurface Singularity ◽  
Milnor Fiber

AbstractWe verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

scholarly journals Real canonical cycle and asymptotics of oscillating integrals

2003 ◽  
Vol 171 ◽  
pp. 187-196
Author(s):  
Daniel Barlet
Keyword(s):  
Analytic Function ◽  
Real Analytic Function ◽  
Isolated Singularity ◽  
Connected Component ◽  
Analytic Set ◽  
Milnor Fiber ◽  
Real Analytic ◽  
Topological Construction

AbstractLet Xℝ ⊂ ℝN a real analytic set such that its complexification Xℂ ⊂ ℂN is normal with an isolated singularity at 0. Let fℝ : Xℝ → ℝ a real analytic function such that its complexification fℂ : Xℂ → ℂ has an isolated singularity at 0 in Xℂ. Assuming an orientation given on to a connected component A of we associate a compact cycle Γ(A) in the Milnor fiber of fℂ which determines completely the poles of the meromorphic extension of or equivalently the asymptotics when T → ±∞ of the oscillating integrals . A topological construction of Γ(A) is given. This completes the results of [BM] paragraph 6.

JACOBIAN IDEALS AND THE NEWTON NON-DEGENERACY CONDITION

2005 ◽  
Vol 48 (1) ◽  
pp. 21-36 ◽  
Author(s):  
Carles Bivià-Ausina
Keyword(s):  
Analytic Function ◽  
Subject Classification ◽  
Function Germ ◽  
Primary 32S05 ◽  
Jacobian Ideal ◽  
Degeneracy Condition ◽  
Łojasiewicz Exponents ◽  
Secondary 57R45

AbstractIn this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Łojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ $f:(\mathbb{C}^n,0)\to(\mathbb{C},0)$. In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.AMS 2000 Mathematics subject classification: Primary 32S05. Secondary 57R45

On the Milnor fiber of non-isolated singularities

2006 ◽  
Vol 43 (1) ◽  
pp. 131-136
Author(s):  
András Horváth ◽  
András Némethi
Keyword(s):  
Homotopy Type ◽  
Singular Locus ◽  
Hypersurface Singularity ◽  
Isolated Singularities ◽  
Milnor Fiber

We prove that the homotopy type of the Milnor fiber Fof a hypersurface singularity f : (Cn+1,0)?(C,0) (n?3) with 1-dimensional singular locus is completely determined by its homology H*(F,Z), and, in fact, it has a canonical bouquet decomposition.

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