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Isolated singular points on complete intersections. (English) Zbl 0552.14002
London Mathematical Society Lecture Note Series, 77. Cambridge etc.: Cambridge University Press. XI, 200 p. £12.50 $ 22.95 (1984).
The book under review is an excellent up-to-date account on the results and methods of the geometry of isolated singular points of complex- analytic spaces, with particular emphasis to the singularities of complete intersections. Before reviewing its contents, let us make a few historical remarks. In 1928 Brauner studied for the first time the singularities of algebraic plane curves from a topological point of view, by associating to such a singularity its link. This link is nothing else than a disjoint (finite) union of circles embedded in \(S^ 3\), in such a way that there is a natural 1-1 correspondence between these circles and the branches of the curve passing through the given singularity. The way in which these circles are knotted or linked in \(S^ 3\) reflects the geometry of the singularity. Later on, in 1961 Mumford extended these ideas to the study of normal 2-dimensional singularities. The link K of such a singularity (which is a differentiable manifold of dimension 3 embedded in a sphere) is studied by using a ”good” resolution of the singularity (X,x). As an application, Mumford found out that K is simply- connected iff x is a smooth point of X. In 1968 Milnor published his famous book ”Singular points of complex hypersurfaces” (Ann. Math. Stud. 61, (1968; Zbl 0184.484), in which he studied the link K of an isolated hypersurface singularity (X,x) by using Morse theory and the Milnor fibration associated to (X,x). In particular, he got a lot of precise and remarkable information concerning the geometry of (X,x). The book under review, written by an expert in singularity theory, not only extends Milnor’s results to the isolated singularities of arbitrary complete intersections, but also gives an overall picture of the up-to-date development of this beautiful theory. The book contains nine chapters, the last two being more advanced. In the first seven chapters the author develops the basic ideas and methods of the theory, such as: the fibration theorem, the Picard Lefschetz formulae, the homotopy type of the Milnor fibre, deformations of isolated singularities, the local monodromy group, a partial classification of isolated singularities of complete intersections. Some of the results of this book belong to its author and did not appear in the literature before. Many of the proofs are new. For example the author gives a new proof of the monodromy theorem (using the techniques of Lê Dũng Tráng). Another example is the following. While Milnor used Morse theory to get informations about the Milnor fibre (in the hypersurface singularity case), the author uses the Picard-Lefschetz formulae in order to do this in the general case. It turns out that these formulae are more suited for the study of deformations of isolated singularities. The last two chapters are more advanced and provide an introduction to the local Gauss-Manin connection and its applications. The first application concerns the period mapping for the miniversal deformation of an isolated complete intersection singularity. One proves that under certain conditions this mapping is a local immersion. In particular, this implies that the Tjurina number is less than or equal to the Milnor number of the singularity. The second application deals with isolated singularities with \({\mathbb{C}}^*\)-action, proving that the Tjurina and the Milnor numbers are equal in case of a singularity with \({\mathbb{C}}^*\)-action which is a complete intersection. As a third application, one investigates the miniversal deformation of a Kleinian singularity.
This book brings together in a very attractive presentation many important results from the theory of isolated singular points, which are due to several mathematicians such as: Arnold, Brieskorn, Greuel, Hamm, Lê Dũng Tráng, the author, Milnor, Steenbrink, Wahl, and many others. The reader is supposed to be acquainted with some prerequisites of algebraic and analytic geometry, algebraic topology and Stein spaces. Given this background, every result stated in the book is proved. Without any doubt, by writing this book the author has done a great service to the mathematical community.
Reviewer: L.Bădescu

14B05 Singularities in algebraic geometry
14M10 Complete intersections
32S05 Local complex singularities
14B07 Deformations of singularities
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry