Milnor Fiber Boundary of a Non-isolated Surface Singularity / by András Némethi, Ágnes Szilárd.

Format:

Book

Author/Creator:

Némethi, András, author.

Szilárd, Ágnes, author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 2037.

Lecture Notes in Mathematics, 0075-8434 ; 2037

Language:

English

Subjects (All):

Differential equations, Partial.

Geometry, Algebraic.

Algebraic topology.

Several Complex Variables and Analytic Spaces.

Algebraic Geometry.

Algebraic Topology.

Local Subjects:

Several Complex Variables and Analytic Spaces.

Algebraic Geometry.

Algebraic Topology.

Physical Description:

1 online resource (XII, 240 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg, 2012.

System Details:

text file PDF

Summary:

In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f,g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f,g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The properties peculiar to M are also emphasized.

Contents:

1 Introduction

2 The topology of a hypersurface germ f in three variables Milnor fiber

3 The topology of a pair (f ; g)

4 Plumbing graphs and oriented plumbed 3-manifolds

5 Cyclic coverings of graphs

6 The graph GC of a pair (f ; g). The definition

7 The graph GC . Properties

8 Examples. Homogeneous singularities

9 Examples. Families associated with plane curve singularities

10 The Main Algorithm

11 Proof of the Main Algorithm

12 The Collapsing Main Algorithm

13 Vertical/horizontal monodromies

14 The algebraic monodromy of H1(¶ F). Starting point

15 The ranks of H1(¶ F) and H1(¶ F nVg) via plumbing

16 The characteristic polynomial of ¶ F via P# and P#

18 The mixed Hodge structure of H1(¶ F)

19 Homogeneous singularities

20 Cylinders of plane curve singularities: f = f 0(x;y)

21 Germs f of type z f 0(x;y)

22 The T;;-family

23 Germs f of type ̃ f (xayb; z). Suspensions

24 Peculiar structures on ¶ F. Topics for future research

25 List of examples

26 List of notations.

Other Format:

Printed edition:

ISBN:

9783642236471

Access Restriction:

Restricted for use by site license.