scholarly journals HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES

2004 ◽  
Vol 15 (09) ◽  
pp. 895-905 ◽  
Author(s):  
W. EBELING ◽  
S. M. GUSEIN-ZADE ◽  
J. SEADE
Keyword(s):  
Complete Intersection ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Analytic Variety ◽  
Curve Singularities ◽  
Complex Analytic Variety ◽  
Arbitrary Curve

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

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.

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

2017 ◽  
Vol 60 (1) ◽  
pp. 175-185 ◽  
Author(s):  
J. J. NUÑO-BALLESTEROS ◽  
B. ORÉFICE-OKAMOTO ◽  
J. N. TOMAZELLA
Keyword(s):  
Sufficient Conditions ◽  
Additional Term ◽  
Milnor Number ◽  
Curve Singularity ◽  
Hypersurface Singularity ◽  
Weighted Degree ◽  
Function Germ ◽  
Analytic Variety ◽  
Complex Analytic Variety ◽  
Necessary And Sufficient

AbstractWe consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

scholarly journals ABOUT THE $\bar{\partial}$-EQUATION AT ISOLATED SINGULARITIES WITH REGULAR EXCEPTIONAL SET

2009 ◽  
Vol 20 (04) ◽  
pp. 459-489 ◽  
Author(s):  
JEAN RUPPENTHAL
Keyword(s):  
Necessary Conditions ◽  
Sufficient Condition ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Exceptional Set ◽  
Analytic Variety ◽  
Stein Domain ◽  
Hölder Estimates

Let Y be a pure dimensional analytic variety in ℂn with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of the present paper is to present a technique which allows us to determine obstructions to the solvability of the [Formula: see text] equation in the L2, respectively L∞, sense on Y* = Y\{0} in terms of certain cohomology classes on X. More precisely, let Ω ⊂⊂ Y be a Stein domain with 0 ∈ Ω, Ω* = Ω\{0}. We give a sufficient condition for the solvability of the [Formula: see text] equation in the L2-sense on Ω*; and in the L∞ sense, if Ω is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hölder estimates for solutions of the [Formula: see text] equation.

scholarly journals The jump of the Milnor number in the X 9 singularity class

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Szymon Brzostowski ◽  
Tadeusz Krasiński
Keyword(s):  
Milnor Number ◽  
Isolated Singularity ◽  
Milnor Numbers

AbstractThe jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.

scholarly journals Goresky–Pardon lifts of Chern classes and associated Tate extensions

2017 ◽  
Vol 153 (7) ◽  
pp. 1349-1371 ◽  
Author(s):  
Eduard Looijenga
Keyword(s):  
Vector Bundle ◽  
Hodge Structure ◽  
Holomorphic Vector Bundle ◽  
Mixed Hodge Structure ◽  
Chern Classes ◽  
Analytic Variety ◽  
Algebraic Setting ◽  
Geometric Conditions ◽  
Stable Cohomology ◽  
Complex Analytic Variety

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

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