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Books > Science & Mathematics > Mathematics > Topology > Algebraic topology
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Normal Surface Singularities (Hardcover, 1st ed. 2022)
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Normal Surface Singularities (Hardcover, 1st ed. 2022)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 74
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This monograph provides a comprehensive introduction to the theory
of complex normal surface singularities, with a special emphasis on
connections to low-dimensional topology. In this way, it unites the
analytic approach with the more recent topological one, combining
their tools and methods. In the first chapters, the book sets out
the foundations of the theory of normal surface singularities. This
includes a comprehensive presentation of the properties of the link
(as an oriented 3-manifold) and of the invariants associated with a
resolution, combined with the structure and special properties of
the line bundles defined on a resolution. A recurring theme is the
comparison of analytic and topological invariants. For example, the
Poincare series of the divisorial filtration is compared to a
topological zeta function associated with the resolution graph, and
the sheaf cohomologies of the line bundles are compared to the
Seiberg-Witten invariants of the link. Equivariant Ehrhart theory
is introduced to establish surgery-additivity formulae of these
invariants, as well as for the regularization procedures of
multivariable series. In addition to recent research, the book also
provides expositions of more classical subjects such as the
classification of plane and cuspidal curves, Milnor fibrations and
smoothing invariants, the local divisor class group, and the
Hilbert-Samuel function. It contains a large number of examples of
key families of germs: rational, elliptic, weighted homogeneous,
superisolated and splice-quotient. It provides concrete
computations of the topological invariants of their links
(Casson(-Walker) and Seiberg-Witten invariants, Turaev torsion) and
of the analytic invariants (geometric genus, Hilbert function of
the divisorial filtration, and the analytic semigroup associated
with the resolution). The book culminates in a discussion of the
topological and analytic lattice cohomologies (as categorifications
of the Seiberg-Witten invariant and of the geometric genus
respectively) and of the graded roots. Several open problems and
conjectures are also formulated. Normal Surface Singularities
provides researchers in algebraic and differential geometry,
singularity theory, complex analysis, and low-dimensional topology
with an invaluable reference on this rich topic, offering a unified
presentation of the major results and approaches.
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