Differential Forms on Singular Varieties - De Rham and Hodge Theory Simplified
A mixed Hodge structure on the cohomology of algebraic varieties was constructed by P. Deligne. For this construction, he used the Hironaka desingularization theorem, Leray spectral sequences, residues for forms with logarithmic singularities and his cohomological descent theory. The main theme of this book is a systematic and elegant theory of differential forms on spaces with singularities. The authors develop a general theory and then use it to give an alternative treatment of mixed Hodge structures, which avoids the use of the cohomological descent theory.
The first part of the book presents the classical Hodge theory on compact Kähler manifolds and the residue theory on a smooth divisor. A summary of properties of differential forms on complex manifolds, sheaf cohomology, the de Rham Laplacian, complex spaces and spectral sequences is included in this first part. The second part of the book describes a systematic theory of differential forms on complex spaces with singularities. Suitable filtrations on the space of forms are defined recursively using induction on dimension. Induced filtrations on the cohomology and the associated spectral sequence then lead to a construction of a mixed Hodge structure on compact spaces. The last part of the book treats mixed Hodge structure on noncompact spaces. It contains a description of the Leray residue theory and its applications in the construction of mixed Hodge structures on noncompact spaces. The book is essentially self-contained and offers a clear, understandable and systematic approach to the theory of mixed Hodge structures. It is a book to be recommended to anybody interested in the topic.