András Némethi: Normal Surface Singularities

Cover: 9783031067525 | Normal Surface Singularities | András Némethi | Buch | xiii | Englisch

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Beschreibung

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.

Über den Autor

András Némethi studied algebraic geometry with Lucian Badescu at Bucharest and then spent 14 years at Ohio State University. He now works at the Alfréd Rényi Institute of Mathematics and at the Eötvös Loránd 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.

Zusammenfassung

Provides a self-contained presentation of analytic and topological invariants of surface singularities

Combines the classical analytic approach with modern low-dimensional topology

Presents lattice cohomologies as categorification of invariants

Inhaltsverzeichnis

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.

Details

Erscheinungsjahr:	2022
Fachbereich:	Arithmetik & Algebra
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Reihe:	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Inhalt:	xiii 722 S.
ISBN-13:	9783031067525
ISBN-10:	3031067525
Sprache:	Englisch
Ausstattung / Beilage:	HC runder Rücken kaschiert
Einband:	Gebunden
Autor:	Némethi, András
Auflage:	1st ed. 2022
Hersteller:	Springer Nature Switzerland Springer International Publishing Springer International Publishing AG Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Maße:	241 x 160 x 45 mm
Von/Mit:	András Némethi
Erscheinungsdatum:	08.10.2022
Gewicht:	1,256 kg

Artikel-ID: 121437074