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Normal Surface Singularities - András Némethi

Normal Surface Singularities

Buch | Hardcover
XIII, 722 Seiten
2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-06752-5 (ISBN)
CHF 82,35 inkl. MwSt
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 Poincaré 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.

Andras Nemethi studied algebraic geometry with Lucian Badescu at Bucharest and then spent 14 years at Ohio State University. He now works at the Alfred Renyi Institute of Mathematics and at the Eoetvoes Lorand University in Budapest. A leading researcher in the theory of complex singularities and their connections with low-dimensional topology, he co-authored the book Milnor Fiber Boundary of a Non-Isolated Surface Singularity, and has authored some 130 research articles, many of them with various collaborators. His honors include an invited address to the International Congress of Mathematicians in 2018. He has built new bridges between analytic and topological invariants (for instance, between the geometric genus and the Seiberg-Witten invariant of the link), proved and formulated several conjectures, and introduced new mathematical objects, such as (topological and analytic) lattice cohomologies and graded roots.

1 Introduction.- 2 Resolution of Surface Singularities.- 3 The Link.- 4 Coverings.- 5 Examples.- 6 Invariants Associated With a Resolution.- 7 The Artin-Laufer Program.- 8  Multivariable Divisorial Filtration.- 9 Topological Invariants. The Seiberg-Witten Invariant.- 10 Ehrhart Theory and the Seiberg-Witten Invariant.- 11 Lattice Cohomology.- 12 Appendix. Complex Analytic Spaces.- References.- Index.

"The book serves its main purpose in a perfect way, and will be a very useful guide or handbook for readers working in algebraic geometry, complex analytic geometry, and algebraic topology. Experts in this field will also enjoy the various topics and discover new perspectives. It is self-contained regarding the main materials and provides detailed explanations of techniques and related topics. For these reasons, this book is a very useful for young researchers as well." (Tomohiro Okuma, zbMATH 1523.14001, 2023)

“The book serves its main purpose in a perfect way, and will be a very useful guide or handbook for readers working in algebraic geometry, complex analytic geometry, and algebraic topology. Experts in this field will also enjoy the various topics and discover new perspectives. It is self-contained regarding the main materials and provides detailed explanations of techniques and related topics. For these reasons, this book is a very useful for young researchers as well.” (Tomohiro Okuma, zbMATH 1523.14001, 2023)

Erscheinungsdatum
Reihe/Serie Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Zusatzinfo XIII, 722 p.
Verlagsort Cham
Sprache englisch
Maße 155 x 235 mm
Gewicht 1262 g
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Geometrie / Topologie
Schlagworte Artin-Laufer program • Casson invariant • Casson-Walker invariant • Kulikov singularities • lattice cohomology • link of singularities • Picard group • Seiberg-Witten invariant • Surgery
ISBN-10 3-031-06752-5 / 3031067525
ISBN-13 978-3-031-06752-5 / 9783031067525
Zustand Neuware
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