Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified uses complexes of differential forms to give a complete treatment of the Deligne theory of mixed Hodge structures on the cohomology of singular spaces. This book features an approach that employs recursive arguments on dimension and does not introduce spaces of higher dimension than the initial space. It simplifies the theory through easily identifiable and well-defined weight filtrations. It also avoids discussion of cohomological descent theory to maintain accessibility. Topics include classical Hodge theory, differential forms on complex spaces, and mixed Hodge structures on noncompact spaces.
“… is divides into three parts of roughly 100 pages each. … written for people who have a basic understanding of the subject and who wish to see what’s happening behind the scenes. … material more accessible. ”
— Donu Arapura, Purdue University, in Siam Review, Vol. 48, No. 3
“This book offers a novel and complete treatment of the Deligne theory of mixed Hodge structures on the cohomology of singular spaces. … well-written for an advanced level text ant it is valuable to beginning researchers as an introduction to an important and beautiful mathematical subject. The book will also serve as a good reference for further research work.”
— Vagn Lundsgaard Hansen (Lyngby), in Zentralblatt MATH 1088
Classical Hodge Theory. Spectral Sequences and Mixed Hodge Structures. Complex Manifolds, Vector Bundles, Differential Forms. Sheaves and Cohomology. Harmonic Forms on Hermitian Manifolds. Hodge Theory on Compact Kählerian Manifolds. The Theory of Residues on a Smooth Divisor. Complex Spaces. Differential Forms on Complex Spaces. The Basic Example. Differential Forms in Complex Spaces. Mixed Hodge Structures on Compact Spaces. Mixed Hodge Structures on Noncompact Spaces. Residues and Hodge Mixed Structures: Leray Theory. Residues and Mixed Hodge Structures on Noncompact Manifolds. Mixed Hodge Structures in Noncompact Spaces: The Basic Example. Mixed Hodge Structures on Noncompact Singular Spaces.