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Singularities and topology of hypersurfaces. (English) Zbl 0753.57001
Universitext. New York etc.: Springer-Verlag. xvi, 263 p. (1992).
This book is neither a graduate text nor a research monograph, but somewhere in between. It expounds how to compute topological invariants of algebraic varieties (mostly hypersurfaces) and their complements — homology groups, fundamental groups, Alexander polynomials, and (eventually mixed) Hodge structures, in both global and local cases, with some emphasis on the relation between the two. Inevitably this requires rather an extensive array of prerequisites. This is dealt with in 3 ways: first, there are three introductory chapters explaining respectively Whitney stratifications, the structure of plane curve and normal surface singularities, and the Milnor fibration and lattice for an isolated hypersurface singularity. As well as giving background material on these topics, numerous examples are included, and calculations of topological invariants giving a foundation for subsequent calculations. Secondly, three appendices give digests of information on integral bilinear forms, weighted projective varieties and mixed Hodge structures respectively. Thirdly, the author is always ready to quote results from other sources and refer the reader to them for further information; modulo this, the book is reasonably self-contained.
The three main chapters of the book describe methods of computation of fundamental groups of hypersurface complements, of cohomology of complete intersections (smooth or with isolated singularities), and of de Rham cohomology of complements of hypersurfaces (with critical locus of dimension \(\leq 1\)). The book is well written: the explanations are firmly rooted in detailed treatments of examples, which indeed occupy most of the text.
This is not a book to skip over: without following how the calculations are done, one loses the whole point. There is a steady progression to more sophisticated ideas, and the final chapter culminates with some very nice results of the author calculating Alexander polynomials, where a crucial ingredient is the defect of some linear system.
It is unusual to find a book devoted to explaining how to make calculations. This is not a book to suit everyone, but for those who want to understand how to calculate topological invariants in local and global complex algebraic geometry it is unrivalled.

MSC:
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
57R19 Algebraic topology on manifolds and differential topology
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32S30 Deformations of complex singularities; vanishing cycles
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
14F25 Classical real and complex (co)homology in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14F40 de Rham cohomology and algebraic geometry
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32S55 Milnor fibration; relations with knot theory
57N80 Stratifications in topological manifolds
32S25 Complex surface and hypersurface singularities
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