Mixed Hodge Structures
Springer Science & Business Media, Feb 27, 2008 - Mathematics - 470 pages
The text of this book has its origins more than twenty- ve years ago. In the seminar of the Dutch Singularity Theory project in 1982 and 1983, the second-named author gave a series of lectures on Mixed Hodge Structures and Singularities, accompanied by a set of hand-written notes. The publication of these notes was prevented by a revolution in the subject due to Morihiko Saito: the introduction of the theory of Mixed Hodge Modules around 1985. Understanding this theory was at the same time of great importance and very hard, due to the fact that it uni es many di erent theories which are quite complicated themselves: algebraic D-modules and perverse sheaves. The present book intends to provide a comprehensive text about Mixed Hodge Theory with a view towards Mixed Hodge Modules. The approach to Hodge theory for singular spaces is due to Navarro and his collaborators, whose results provide stronger vanishing results than Deligne’s original theory. Navarro and Guill en also lled a gap in the proof that the weight ltration on the nearby cohomology is the right one. In that sense the present book corrects and completes the second-named author’s thesis.
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abelian category algebraic variety canonical cohomology groups compactification complex algebraic variety complex manifold complex of sheaves components cone constant sheaf construction Corollary cup product cycle defined definition degenerates Deligne denote derived category differential graded algebra dimension direct image divisor duality DX-module Example fibration fibre finite follows functor hence Hodge conjecture Hodge filtration Hodge theory holomorphic holonomic homology hypercohomology inclusion induces injective intersection irreducible isomorphism iterated integrals Kähler manifold Lefschetz Lemma Leray spectral sequence Let f long exact sequence Math minimal model mixed Hodge complexes mixed Hodge modules mixed Hodge structures monodromy morphism f morphism of mixed Proof pure Hodge structure quasi-isomorphism quotient R-Hodge rational Remark resolution respectively Rham complex semi-simplicial simple normal crossings singular structure of weight subset subspace subvariety surjective tensor Theorem topological space trivial vanishing variation of Hodge vector bundle vector space weight filtration zero