Milnor Fiber Boundary of a Non-isolated Surface Singularity, 2012 Lecture Notes in Mathematics Series, Vol. 2037

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.

Presents a new approach in the study of non-isolated hypersurface singularities The first book about non-isolated hypersurface singularities Conceptual and comprehensive description of invariants of non-isolated singularities Key connections between singularity theory and low-dimensional topology Numerous explicit examples for plumbing representation of the boundary of the Milnor fiber Numerous explicit examples for the Jordan block structure of different monodromy operators