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Mixed Hodge structures and singularities. (English) Zbl 0902.14005

Cambridge Tracts in Mathematics. 132. Cambridge: Cambridge University Press. xxi, 186 p. (1998).
According to the publication standards of Cambridge Tracts in Mathematics, the present book provides a thorough yet reasonably concise treatment of a specific topic by taking up a single thread in a wide subject, following various ramifications of it, and illuminating thus its different aspects from a central point of view. The topic of the treatise under review is the local theory of singularities of algebraic varieties, analytic spaces, or morphisms, and the thread taken up is the investigation of singularities via the cohomology theory for sheaves of differential forms, i.e., by means of methods of Hodge theory and its diverse generalizations.
Thus, in a body, this book provides both an introduction to, and a survey of, some central aspects of singularity theory, such as they have been intensively studied over the past thirty years. The text consists of three chapters, each of which is subdivided into several sections. To chapter I, which is preceded by a very thorough, detailed, motivating and masterly written introduction to the contents of the book, has been given the title “The Gauss-Manin connection”. Here the author explains, always in a very concise but comprehensive and lucid manner, many of the fundamental ideas, methods, techniques, and results centred around this crucial concept. This includes: Milnor fibration, Picard-Lefschetz monodromy transformations, locally constant sheaves and systems of linear differential equations, integrable connections, relative De Rham cohomology sheaves, meromorphic connections, Brieskorn lattices, quasi-homogeneous singularities, singular points of systems of linear differential equations, regularity of the Gauss-Manin connection, period matrices, the geometry of the Picard-Fuchs equation, the monodromy theorem, the Gauss-Manin connection in the case of non-isolated hypersurface singularities, and many other related topics.
Chapter II deals with the various kinds of Hodge structures and their variation behavior under deformations. This chapter is entitled “Limit mixed Hodge structure on the vanishing cohomology of an isolated hypersurface singularity”. Here some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and their associated period maps, are developed in the local situation of isolated singularities of holomorphic functions. The main topics of this chapter are, among others, the following ones: mixed Hodge structures, polarized Hodge structures, Deligne’s theorem, limit Hodge structures in the sense of W. Schmid, limit mixed Hodge structures in the sense of J. Steenbrink, the Hodge theory of smooth hypersurfaces (after Griffiths-Deligne), the Gauss-Manin system of an isolated singularity, decompositions of meromorphic connections, the limit Hodge filtration due to Varchenko and Scherk-Steenbrink, and spectra of various types of singularities.
The concluding chapter III, entitled “The period map of a Milnor-constant deformation of an isolated hypersurface singularity associated with Brieskorn lattices and mixed Hodge structures”, discusses the glueing of Milnor fibrations, meromorphic connections of Milnor-constant deformations of singularities, root components of geometric sections, the period map for embeddings of Brieskorn lattices and various types of singularities, the period map associated with the mixed Hodge structure on the vanishing cohomology, the infinitesimal Torelli theorem, the period map for quasi-homogeneous singularities, the Picard-Fuchs singularity, and the recent results of C. Hertling for hypersurface singularities.
Without any doubt, the author has covered a wealth of material on a highly advanced topic in complex geometry, and in this regard he has provided a great service to the mathematical community, first and foremost with a view to the systematic, comprehensive understanding of the vast realm of singularity theory. Although he had to relinquish almost all proofs of the numerous deep theorems, he has succeeded in providing a brilliant introduction to, and a comprehensive overview of, this contemporary central subject of complex geometry. This book is designed for researchers whose interests are closely bound up with singularity theory, algebraic geometry, and complex analysis, and in that it is an excellent source and guide for them. Also, the entire text represents an irresistible invitation to the subject, and may be seen as a dependable pathfinder with regard to the vast existing original literature in the field.

MSC:

14B05 Singularities in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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
32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities