Handbook of Geometry and Topology of Singularities I

This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs (C, o) of complex analytic curves contained in a smooth complex analytic surface S. The embedded topological type of such a pair (S, C) is usually defined to be that of the oriented link obtained by intersecting C with a sufficiently small oriented Euclidean sphere centered at the point o, defined once a system of local coordinates (x, y) was chosen on the germ (S, o). If one works more generally over an arbitrary algebraically closed field of characteristic zero, one speaks instead of the combinatorial type of (S, C). One may define it by looking either at the Newton-Puiseux series associated to C relative to a generic local coordinate system (x, y), or at the set of infinitely near points which have to be blown up in order to get the minimal embedded resolution of the germ (C, o) or, thirdly, at the preimage of this germ by the resolution. Each point of view leads to a different encoding of the combinatorial type by a decorated tree: an Eggers-Wall tree, an Enriques diagram, or a weighted dual graph. The three trees contain the same information, which in the complex setting is equivalent to the knowledge of the embedded topological type. There are known algorithms for transforming one tree into another. In this paper we explain how a special type of two-dimensional simplicial complex called a lotus allows to think geometrically about the relations between the three types of trees. Namely, all of them embed in a natural lotus, their numerical decorations appearing as invariants of it. This lotus is constructed from the finite set of Newton polygons created during any process of resolution of (C, o) by successive toric modifications.

We consider a reduced complex surface germ (X, p). We do not assume that X is normal at p, and so, the singular locus ( Σ, p) of (X, p) could be one dimensional. This text is devoted to the description of the topology of (X, p). By the conic structure theorem (see Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematical Studies 61 (1968), Princeton Univ. Press), (X, p) is homeomorphic to the cone on its link L _X. First of all, for any good resolution ρ : (Y, E _Y) → (X, 0) of (X, p), there exists a factorization through the normalization \(\nu : (\bar X,\bar p) \to (X,0 )\) (see H. Laufer, Normal two dimensional singularities, Ann. of Math. Studies 71, (1971), Princeton Univ. Press., Thm. 3.14). This is why we proceed in two steps.

In Sect. 2.4, we give a detailed proof of the existence of a good resolution of a normal surface germ by the Hirzebruch-Jung method (Theorem 2.4.6). With this method a good resolution is obtained via an embedded resolution of the discriminant of π (see Friedrich Hirzebruch, Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen, Math. Ann. 126 (1953) p. 1–22). An example is given Sect. 2.6. An appendix (Sect. 2.5) is devoted to the topological study of lens spaces and to the description of the minimal resolution of quasi-ordinary singularities of surfaces. Section 2.5 provides the necessary background material to make the proof of Theorem 2.4.6 self-contained.

The problem of resolution of singularities and its solution in various contexts can be traced back to I. Newton and B. Riemann. This paper is an attempt to give a survey of the subject starting with Newton till the modern times, as well as to discuss some of the main open problems that remain to be solved. The main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski’s approach via valuations, Hironaka’s celebrated result in characteristic zero and all dimensions and its subsequent strengthenings and simplifications, existing results in positive characteristic (mostly up to dimension three), de Jong’s approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singularities of vector fields and foliations. In many places, we have tried to summarize the main ideas of proofs of various results without getting too much into technical details.

This is a survey of stratification theory in the differentiable category from its beginnings with Whitney, Thom and Mather until the present day. We concentrate mainly on the properties of C ^∞ stratified sets and of stratifications of subanalytic or definable sets, with some reference to stratifications of complex analytic sets. Brief mention is made of the theory of stratified mappings.

After the local topological structure of stratified spaces was determined by R. Thom (Bull. Amer. Math. Soc., 75 (1969), 240–284) and J. Mather (Notes on topological stability, lecture notes, Harvard University, 1970) it became possible (see Kashiwara and Schapira, Sheaves on Manifolds, Grundlehren der math. Wiss. 292, Springer Verlag Berlin, Heidelberg, 1990; Goresky and MacPherson, Stratified Morse Theory, Ergebnisse Math. 14, Springer Verlag, Berlin, Heidelberg, 1988; Schürmann, Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne 63, Birkhäuser Verlag, Basel, 2003) to analyze constructible sheaves on a stratified space using Morse theory. Although the detailed proofs are formidable, the statements and main ideas are simple and intuitive. This article is a survey of the constructions and results surrounding this circle of ideas.

The fibration theorem for analytic maps near a critical point published by John Milnor in 1968 is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic “folklore theorems” which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity \((X, \underline {0})\) does not depend on the choice of functions that define it. We glance at the use of polar varieties to studying the topology of singularities, which springs from ideas by René Thom. We give an elementary proof of a fundamental “attaching-handles” theorem, which is key for describing the topology of the Milnor fibers. This is also related to the so-called “carousel”, that allows a deeper understanding of the topology of plane curves and has several applications in various settings. Finally we speak about Lê’s conjecture concerning map-germs , and about the Lê-Ramanujam theorem, which still is open in dimension 2.

We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.

We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of monodromy.

In these notes we consider different theorems of the Lefschetz type. We start with the classical Lefschetz Theorem for hyperplane sections on a non-singular projective variety. We show that this extends to the cases of a non-singular quasi-projective variety and to singular varieties. We also consider local forms of theorems of the Lefschetz type.

Complex simple Lie algebras with simply laced root systems are classified by Dynkin diagrams of type A_n, D_n, E₆, E₇, and E₈. Also the dual graphs of the minimal resolution of Kleinian singularities are precisely the same aforementioned Dynkin diagrams. In this work, we recall the basic definitions and some results of the theory of complex Lie algebras and of Kleinian singularities, in order to present a relation between finite dimensional complex simple Lie algebras and the Kleinian singularities, given by a theorem by Brieskorn. We also present the extension of Brieskorn’s theorem to the simple elliptic singularity \(\tilde {D}_5\).