Normal Surface Singularities

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.

本論文全面介紹了複雜正規曲面奇異點理論，特別強調其與低維拓撲的聯繫。以此方式，將分析方法與最近的拓撲方法結合，結合它們的工具和方法。

在前幾章中，本書建立了正規曲面奇異點理論的基礎。包括對鏈結（作為一個定向的三維流形）的性質和與解析的關聯的全面介紹，以及在解析中定義的線束的結構和特殊性質。一個重要的主題是比較解析和拓撲的不變量。例如，將除子過濾的庞加莱級數與解析圖相關的拓撲ζ函數進行比較，將線束的層上同調與鏈結的 Seiberg-Witten 不變量進行比較。引入等變 Ehrhart 理論以建立這些不變量的手術可加性公式，以及多變量級數的正規化程序。

除了最近的研究，本書還介紹了更經典的主題，如平面和尖點曲線的分類，Milnor 纖維和平滑不變量，局部除子類群和 Hilbert-Samuel 函數。它包含了許多關鍵族群的示例：有理、橢圓、加權齊次、超隔離和拼接商。它提供了這些鏈結的拓撲不變量（Casson(-Walker) 和 Seiberg-Witten 不變量，Turaev 扭曲）和解析不變量（幾何屬性、除子過濾的 Hilbert 函數和與解析相關的半群）的具體計算。本書以拓撲和解析格上同調（作為 Seiberg-Witten 不變量和幾何屬性的分類）以及分級根的討論作為結尾。還提出了幾個未解決的問題和猜想。

《正規曲面奇異點》為代數和微分幾何、奇異點理論、復分析和低維拓撲的研究人員提供了一個寶貴的參考，提供了這一豐富主題的統一介紹，並呈現了主要結果和方法。

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.

András Némethi在布加勒斯特跟隨Lucian Badescu學習代數幾何，然後在俄亥俄州立大學度過了14年。他現在在布達佩斯的阿爾弗雷德·雷尼數學研究所和埃特沃什·洛蘭德大學工作。作為複雜奇異性理論及其與低維拓撲的聯繫的領先研究者，他與他人合著了《非孤立曲面奇異性的Milnor纖維邊界》一書，並撰寫了約130篇研究文章，其中許多與不同的合作者合作。他的榮譽包括2018年國際數學家大會的邀請演講。他在分析和拓撲不變量之間建立了新的橋樑（例如，幾何屬性和鏈結的Seiberg-Witten不變量之間），證明和提出了幾個猜想，並引入了新的數學對象，例如（拓撲和分析的）格子上同調和分級根。