Mixed Hodge Structures and Singularities by Valentine S. Kulikov (PDF)

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This vital work is both an introduction to, and a survey of singularity theory, in particular, studying singularities by means of differential forms. Here, some ideas and notions that arose in global algebraic geometry, namely mixed Hodge structures and the theory of period maps, are developed in the local situation to study the case of isolated singularities of holomorphic functions. The author introduces the Gauss-Manin connection on the vanishing cohomology of a singularity, that is on the cohomology fibration associated to the Milnor fibration, and draws on the work of Brieskorn and Steenbrink to calculate this connection, and the limit mixed Hodge structure. This is an excellent resource for all researchers in singularity theory, algebraic or differential geometry.

User’s Reviews

Product description Review ‘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 … he has succeeded in providing a brilliant introduction to, and a comprehensive overview of, this contemporary central subject of complex geometry … 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.’ W. Kleinert, Zentralblatt für Mathematik und ihre Grenzgebiete Book Description This 1998 book is a modern introduction to, and a survey of, some topics of singularity theory.

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