Milnor fiber boundary of a non-isolated surface singularity
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Bibliographic Information
Milnor fiber boundary of a non-isolated surface singularity
（Lecture notes in mathematics, 2037）
Springer, c2012
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||2037200021327314
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Note
Includes bibliographical references (p. 231-236) and index
Description and Table of Contents
Description
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.
Table of 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
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