Mixed Hodge Structures And Singularities

Mixed Hodge Structures and Singularities

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.

Mixed Hodge Structures

This is comprehensive basic monograph on mixed Hodge structures. Building up from basic Hodge theory the book explains Delingne's mixed Hodge theory in a detailed fashion. Then both Hain's and Morgan's approaches to mixed Hodge theory related to homotopy theory are sketched. Next comes the relative theory, and then the all encompassing theory of mixed Hodge modules. The book is interlaced with chapters containing applications. Three large appendices complete the book.

Mixed Hodge Structures and Singularities