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

Némethi, András. author., Author,
Berlin, Heidelberg : Springer Berlin Heidelberg, 2012.
Lecture Notes in Mathematics, 0075-8434 ; 2037
Lecture Notes in Mathematics, 0075-8434 ; 2037
1 online resource (XII, 240 pages)
Differential equations, partial.
Geometry, algebraic.
Algebraic topology.
Local subjects:
Several Complex Variables and Analytic Spaces. (search)
Algebraic Geometry. (search)
Algebraic Topology. (search)
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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.
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.
Szilárd, Ágnes. author., Author,
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10.1007/978-3-642-23647-1 doi
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