This book presents basic results from commutative algebra, differential topology, several complex variables, regular solids with applications to the isolated singularities and is based on a course given by the author at the University of Cologne. The book has five chapters. In the first chapter (Regular solids and finite rotation groups) the general concepts of convex polytopes and their regularity are introduced and the connection between the regular solids and the finite groups of rotations is established. The classification of the finite subgroups of SO(3) is given.
In the chapter II (Finite subgroups of SL(2,\({\mathbb{C}})\) and invariant polynomials) a relation between the finite subgroups \(\Gamma\) of SO(3) and the finite subgroups G of the group SL(2,\({\mathbb{C}})\) is firstly established; then the convex hull of G in \({\mathbb{C}}^ 2={\mathbb{R}}^ 4\) is studied. The action of G on \({\mathbb{C}}^ 2\) is studied using the geometry of the Platonic solids and a description of the \({\mathbb{C}}\)-algebra \(S^ G\) of G-invariant polynomials is given.
The chapter III (Local theory of several complex variables) is a concise introduction to the domain with some details of the local structure of finite maps.
In the chapter IV (Quotient singularities and their resolutions) the quotient singularities (V,0) of the finite subgroups of SL(2,\({\mathbb{C}})\) are studied. Let \(\tilde V\to^{f}V\) be the resolution of the singularity and let \(E=f^{-1}(0)\) be the exceptional set. The topology of \(E\subset \tilde V\) is related to the presentation of the finite subgroups \(G\subset SL(2,{\mathbb{C}})\) by generators and relations and to the 3-manifolds \(S^ 3/G.\)
The chapter V (The hierarchy of simple singularities) provides the beginning of the classification of the germs of holomorphic functions in three variables up to holomorphic equivalence: The equivalence classes of simple germs belong to two infinite series \(A_ k\) \((k=0,1,...)\) and \(D_ k\) \((k=4,5..)\) and to one finite series \(E_ k\) \((k=6,7,8)\). A germ f in three variables is simple if and only if its zero set N(f) is isomorphic to the germ of the quotient variety \({\mathbb{C}}^ 2/G\) of a finite subgroup \(G\subset SL(2,{\mathbb{C}})\). The two infinite series \(A_ k\) and \(D_ k\) correspond to the cyclic and binary dihedral groups respectively, and \(E_ 6\), \(E_ 7\), \(E_ 8\) correspond to the binary tetrahedral, octahedral and icosahedral groups.
The book is written in a very clear style and it is understandable for the non-experts.