Mixed Hodge Structures (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 52)

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. Basic Hodge Theory -- Compact Kähler Manifolds -- Pure Hodge Structures -- Abstract Aspects Of Mixed Hodge Structures -- Mixed Hodge Structures On Cohomology Groups -- Smooth Varieties -- Singular Varieties -- Singular Varieties: Complementary Results -- Applications To Algebraic Cycles And To Singularities -- Mixed Hodge Structures On Homotopy Groups -- Hodge Theory And Iterated Integrals -- Hodge Theory And Minimal Models -- Hodge Structures And Local Systems -- Variations Of Hodge Structure -- Degenerations Of Hodge Structures -- Applications Of Asymptotic Hodge Theory -- Perverse Sheaves And D-modules -- Mixed Hodge Modules. Chris A. M. Peters, Joseph H. M. Steenbrink. Includes Bibliographical References (p. [445]-456) And Indexes. 158320_1_En_FM1_Chapter_OnlinePDF.pdf 158320_1_En_1_Chapter_OnlinePDF.pdf 158320_1_En_1_PartFrontmatter_OnlinePDF.pdf 158320_1_En_2_Chapter_OnlinePDF.pdf 158320_1_En_3_Chapter_OnlinePDF.pdf 158320_1_En_4_Chapter_OnlinePDF.pdf 158320_1_En_2_PartFrontmatter_OnlinePDF.pdf 158320_1_En_5_Chapter_OnlinePDF.pdf 158320_1_En_6_Chapter_OnlinePDF.pdf 158320_1_En_7_Chapter_OnlinePDF.pdf 158320_1_En_8_Chapter_OnlinePDF.pdf 158320_1_En_3_PartFrontmatter_OnlinePDF.pdf 158320_1_En_9_Chapter_OnlinePDF.pdf 158320_1_En_10_Chapter_OnlinePDF.pdf 158320_1_En_4_PartFrontmatter_OnlinePDF.pdf 158320_1_En_11_Chapter_OnlinePDF.pdf 158320_1_En_12_Chapter_OnlinePDF.pdf 158320_1_En_13_Chapter_OnlinePDF.pdf 158320_1_En_14_Chapter_OnlinePDF.pdf 158320_1_En_15_Chapter_OnlinePDF.pdf 158320_1_En_BM1_Chapter_OnlinePDF.pdf This is the first comprehensive basic monograph on mixed Hodge structures. Starting with a summary of classic Hodge theory from a modern vantage point the book goes on to explain Deligne's mixed Hodge theory. Here proofs are given using cubical schemes rather than simplicial schemes. Next come Hain's and Morgan's results on mixed Hodge structures related to homotopy theory. Steenbrink's approach of the limit mixed Hodge structure is then explained using the language of nearby and vanishing cycle functors bridging the passage to Saito's theory of mixed Hodge modules which is the subject of the