Milnor Fiber Boundary of a Non-Isolated Surface Singularity

ISBN-13: 9783642236464 / Angielski / Miękka / 2012 / 240 str.

Andr?'s N. Methi; Gnes Szil Rd

Opis

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 emphasize

Recenzje

From the reviews:

"The aim of this book is to study the topological types of the oriented smooth 3-manifolds appearing as boundaries F of the Milnor fibers of complex surface singularities of embedding dimension 3, as well as the monodromy actions on their homology. ... It is clearly invaluable for anybody interested in the topology of non-isolated complex surface singularities and even of singularities of real analytic spaces of dimension 4." (Patrick Popescu-Pampu, Mathematical Reviews, January, 2014)

"The book describes three manifolds which occur in relation with complex hypersurfaces in C³ near singular points. ... I recommend it to all students and researchers who are interested in the local topology of algebraic varieties. It contains a good description of techniques, such as plumbing, cyclic coverings, monodromy, et cetera. The book is well written and ends with several topics for future research." (Dirk Siersma, Nieuw Archief voor Wiskunde, Vol. 14 (2), June, 2013)

Spis treści

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

Blurb

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