Milnor Fiber Boundary of a Non-isolated Surface Singularity

1. Auflage 2012
ISBN: 978-3-642-23646-4
Verlag: Springer

Buch, Englisch, Band 2037, 240 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 394 g

Reihe: Lecture Notes in Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Némethi, András

Szilárd, Ágnes

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

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 G C of a pair ( f ; g ). The definition.- 7 The graph G C . Properties.- 8 Examples. Homogeneous singularities.- 9 Examples. Families associated with plane curve singularities.- 10 The Main Algorithm.- 11 Proof of the Main Algorithm.- 12The Collapsing Main Algorithm.- 13 Vertical/horizontal monodromies.- 14 The algebraic monodromy of H1(¶ F). Starting point.- 15 The ranks of H 1(¶ F ) and H 1(¶ F nVg ) via plumbing.- 16 The characteristic polynomial of ¶ F via P # and P #.- 18 The mixed Hodge structure of H 1(¶ 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