Milnor number and classification of isolated singularities of holomorphic maps

1982 ◽  
pp. 1-34 ◽  
Author(s):  
Bruce Bennett ◽  
Stephen S. -T. Yau
Keyword(s):  
Milnor Number ◽  
Holomorphic Maps ◽  
Isolated Singularities

HERMITIAN COMPLEXITY OF REAL POLYNOMIAL IDEALS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250026 ◽  
Author(s):  
JOHN P. D'ANGELO ◽  
MIHAI PUTINAR
Keyword(s):  
Necessary Conditions ◽  
Complex Geometry ◽  
Polynomial Optimization ◽  
Symmetric Polynomial ◽  
Holomorphic Maps ◽  
Real Polynomial ◽  
Algebraic Subset ◽  
Algebraic Set ◽  
Holomorphic Embedding

We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of Cn. This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls.

scholarly journals Milnor invariants of clover links

2016 ◽  
Vol 27 (13) ◽  
pp. 1650108
Author(s):  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Repeated Sequence ◽  
Milnor Number ◽  
Greatest Common Divisor ◽  
Homotopy Classification ◽  
Number Formula ◽  
Milnor Numbers ◽  
Milnor Invariants

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to [Formula: see text] are well-defined if those of length [Formula: see text] vanish, and that Milnor numbers of length at least [Formula: see text] are not well-defined if those of length [Formula: see text] survive. For a clover link [Formula: see text] with vanishing Milnor numbers of length [Formula: see text], we show that the Milnor number [Formula: see text] for a sequence [Formula: see text] is well-defined by taking modulo the greatest common divisor of the [Formula: see text], where [Formula: see text] is any proper subsequence of [Formula: see text] obtained by removing at least [Formula: see text] indices. Moreover, if [Formula: see text] is a non-repeated sequence of length [Formula: see text], the possible range of [Formula: see text] is given explicitly. As an application, we give an edge-homotopy classification of [Formula: see text]-clover links.

Stable mappings of discriminant varieties

1988 ◽  
Vol 103 (1) ◽  
pp. 69-82
Author(s):  
J. W. Bruce
Keyword(s):  
Wave Front ◽  
Smooth Surfaces ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Low Dimensions

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.

scholarly journals A Classification of Isolated Singularities of Elliptic Monge-Ampére Equations in Dimension Two

2015 ◽  
Vol 68 (12) ◽  
pp. 2085-2107 ◽  
Author(s):  
José A. Gálvez ◽  
Asun Jiménez ◽  
Pablo Mira
Keyword(s):  
Isolated Singularities ◽  
Monge Ampere

scholarly journals Codimension Two Determinantal Varieties with Isolated Singularities

2014 ◽  
Vol 115 (2) ◽  
pp. 161 ◽  
Author(s):  
Maria Aparecida Soares Ruas ◽  
Miriam Da Silva Pereira
Keyword(s):  
Betti Number ◽  
Normal Forms ◽  
Milnor Number ◽  
Isolated Singularities ◽  
Generic Fiber ◽  
Linear Projection ◽  
Codimension Two ◽  
Determinantal Varieties ◽  
Surface Singularities ◽  
Generic Section

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in $\mathsf{C}^4$, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

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