ebook ebooks e-book e-books downloaden bei MyEbooks.ch downloaden

Mixed Hodge Structures

:	Christiaan Peters, Joseph H. M. Steenbrink
:	Mixed Hodge Structures
:	Springer-Verlag
:	9783540770176
:	1
:	CHF 124.70
:

:	Arithmetik, Algebra
:	English

:	470
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	Preface	5
	Contents	6
	Introduction	13
	Part I Basic Hodge Theory	21
	1 Compact K ahler Manifolds	22
	1.1 Classical Hodge Theory	22
	1.2 The Lefschetz Decomposition	31
	1.3 Applications	39
	2 Pure Hodge Structures	44
	2.1 Hodge Structures	44
	2.2 Mumford-Tate Groups of Hodge Structures	51
	2.3 Hodge Filtration and Hodge Complexes	54
	2.4 Re ned Fundamental Classes	62
	2.5 Almost K ahler V -Manifolds	67
	3 Abstract Aspects of Mixed Hodge Structures	72
	3.1 Introduction to Mixed Hodge Structures: Formal Aspects	73
	3.2 Comparison of Filtrations	77
	3.3 Mixed Hodge Structures and Mixed Hodge Complexes	80
	3.4 The Mixed Cone	87
	3.5 Extensions of Mixed Hodge Structures	90
	Part II Mixed Hodge structures on Cohomology Groups	97
	4 Smooth Varieties	98
	4.1 Main Result	98
	4.2 Residue Maps	101
	4.3 Associated Mixed Hodge Complexes of Sheaves	105
	4.4 Logarithmic Structures	108
	4.5 Independence of the Compacti cation and Further Complements	110
	5 Singular Varieties	118
	5.1 Simplicial and Cubical Sets	118
	5.2 Construction of Cubical Hyperresolutions	128
	5.3 Mixed Hodge Theory for Singular Varieties	133
	5.4 Cup Product and the K unneth Formula.	142
	5.5 Relative Cohomology	144
	6 Singular Varieties: Complementary Results	149
	6.1 The Leray Filtration	149
	6.2 Deleted Neighbourhoods of Algebraic Sets	152
	6.3 Cup and Cap Products, and Duality	160
	7 Applications to Algebraic Cycles and to Singularities	168
	7.1 The Hodge Conjectures	168
	7.2 Deligne Cohomology	175
	7.3 The Filtered De Rham Complex And Applications	180
	Part III Mixed Hodge Structures on Homotopy Groups	195
	8 Hodge Theory and Iterated Integrals	196
	8.1 Some Basic Results from Homotopy Theory	197
	8.2 Formulation of the Main Results	201
	8.3 Loop Space Cohomology and the Homotopy De Rham Theorem	204
	8.4 The Homotopy De Rham Theorem for the Fundamental Group	210
	8.5 Mixed Hodge Structure on the Fundamental Group	213
	8.6 The Sullivan Construction	216
	8.7 Mixed Hodge Structures on the Higher Homotopy Groups	218
	9 Hodge Theory and Minimal Models	223
	9.1 Minimal Models of Di erential Graded Algebras	224
	9.2 Postnikov Towers and Minimal Models	the Simply Connected Case
	9.3 Mixed Hodge Structures on the Minimal Model	228
	9.4 Formality of Compact K ahler Manifolds	234
	Part IV Hodge Structures and Local Systems	241
	10 Variations of Hodge Structure	242
	10.1 Preliminaries: Local Systems over Complex Manifolds	242
	10.2 Abstract Variations of Hodge Structure	244
	10.3 Big Monodromy Groups, an Application	248
	10.4 Variations of Hodge Structures Coming From Smooth Families	251
	11 Degenerations of Hodge Structures	256
	11.1 Local Systems Acquiring Singularities	256
	11.2 The Limit Mixed Hodge Structure on Nearby Cycle Spaces	262
	11.3 Geometric Consequences for Degenerations	277
	11.4 Examples	288
	12 Applications of Asymptotic Hodge theory	291
	12.1 Applications to Singularities	291
	12.2 An Application to Cycles: Grothendieck's Induction Principle	297
	13 Perverse Sheaves and D-Modules	302
	13.1 Verdier Duality	302
	13.2 Perverse Complexes	307
	13.3 Introduction to D-Modules	314
	13.4 Coherent D-Modules	321
	13.5 Filtered D-modules	328
	13.6 Holonomic D-Modules	330
	14 Mixed Hodge Modules	338
	14.1 An Axiomatic Introduction	339
	14.2 The Kashiwara-Malgrange Filtration	348
	14.3 Polarizable Hodge Modules	354
	14.4 Mixed Hodge Modules	363
	Part V Appendices	373
	A Homological Algebra	374
	A.1 Additive and Abelian Categories	374
	A.2 Derived Categories	379
	A.3 Spectral Sequences and Filtrations	393
	B Algebraic and Di erential Topology	403
	B.1 Singular (Co)homology and Borel-Moore Homology	403
	B.2 Sheaf Cohomology	408
	B.3 Local Systems and Their Cohomology	425
	C Strati ed Spaces and Singularities	431
	C.1 Strati ed Spaces	431
	C.2 Fibrations, and the Topology of Singularities	435
	References	443
	Index of Notations	455
	Index	458