Differential forms on singular varieties: De Rham and Hodge theory simplified
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, FL [u.a.]
Chapman & Hall/CRC
2006
|
| Series: | Monographs and textbooks in pure and applied mathematics
273 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 307-[310]) and index |
| Physical Description: | xix, 311 p. graph. Darst. 24 cm |
| ISBN: | 0849337399 9780849337390 |
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MARC
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| 100 | 1 | |a Ancona, Vincenzo |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential forms on singular varieties |b De Rham and Hodge theory simplified |c Vincenzo Ancona ; Bernard Gaveau |
| 264 | 1 | |a Boca Raton, FL [u.a.] |b Chapman & Hall/CRC |c 2006 | |
| 300 | |a xix, 311 p. |b graph. Darst. |c 24 cm | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Monographs and textbooks in pure and applied mathematics |v 273 | |
| 500 | |a Includes bibliographical references (p. 307-[310]) and index | ||
| 650 | 4 | |a Formes différentielles | |
| 650 | 4 | |a Géométrie algébrique | |
| 650 | 4 | |a Hodge, Théorie de | |
| 650 | 4 | |a Singularités (Mathématiques) | |
| 650 | 4 | |a Differential forms | |
| 650 | 4 | |a Hodge theory | |
| 650 | 4 | |a Singularities (Mathematics) | |
| 650 | 4 | |a Geometry, Algebraic | |
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| 650 | 0 | 7 | |a Hodge-Theorie |0 (DE-588)4135967-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kähler-Mannigfaltigkeit |0 (DE-588)4162978-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a DeRham-Kohomologie |0 (DE-588)4352640-8 |2 gnd |9 rswk-swf |
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| _version_ | 1804135239962853376 |
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| adam_text | DIFFERENTIAL FORMS ON SINGULAR VARIETIES DE RHAM AND HODGE THEORY
SIMPLIFIED VINCENZO ANCONA UNIVERSITY OFFIRENZE FLORENCE, ITALY BERNARD
GAVEAU UNIVERSITY PIERRE ET MARIE CURIE PARIS, FRANCE CHAPMAN & HALL/CRC
TAYLOR &. FRANCIS GROUP BOCA RATON LONDON NEW YORK SINGAPORE CONTENTS I
CLASSICAL HODGE THEORY 1 1 SPECTRAL SEQUENCES AND MIXED HODGE STRUCTURES
3 1.1 INTRODUCTION 3 1.2 FILTRATIONS 3 1.3 STRICT MORPHISMS 4 1.4
FILTERED COMPLEXES 5 1.5 SPECTRAL SEQUENCES 5 1.6 THE FIRST TERM OF THE
SPECTRAL SEQUENCE 6 1.7 THE GRADED COHOMOLOGY 7 1.8 PURE HODGE
STRUCTURES 8 1.9 MORPHISMS OF PURE HODGE STRUCTURES 9 1.10 MIXED HODGE
STRUCTURES 11 1.11 EXACT SEQUENCES OF MIXED HODGE STRUCTURES 14 1.12
SHIFTED COMPLEXES AND SHIFTED FILTRATIONS 15 1.13 THE STRICTNESS OF D
AND THE DEGENERATION OF THE SPECTRAL SE- QUENCE 15 1.14 FLAT MODULES 16
1.15 CONNECTING HOMOMORPHISMS 17 2 COMPLEX MANIFOLDS, VECTOR BUNDLES,
DIFFERENTIAL FORMS 19 2.1 INTRODUCTION 19 2.2 COMPLEX MANIFOLDS 19 2.2.1
DEFINITIONS OF COMPLEX COORDINATES AND MANIFOLDS ... 19 2.2.2 TANGENT
VECTORS 20 2.2.3 HOLOMORPHIC FUNCTIONS . 22 2.2.4 COMPLEX SUBMANIFOLDS
22 2.2.5 EXAMPLES 23 2.3 COMPLEX VECTOR BUNDLES AND DIVISORS 25 2.3.1
OPERATIONS ON BUNDLES . 27 2.3.2 TANGENT BUNDLE 28 2.3.3 EXAMPLE:
COMPLEX TORI 29 2.3.4 LINE BUNDLES AND DIVISORS 29 2.3.5 EXAMPLE: P* AND
ITS LINE BUNDLES 30 2.4 DIFFERENTIAL FORMS ON COMPLEX MANIFOLDS . . . 31
2.4.1 EXPRESSIONS IN LOCAL COORDINATES 32 XM XIV 2.4.2 THE HODGE
FILTRATIONS F AND F 33 2.4.3 PULLBACK 33 2.4.4 EXTERIOR DIFFERENTIALS 35
2.4.5 EXTERIOR DIFFERENTIALS AND PULLBACK 36 2.4.6 DIFFERENTIALS AND
EXTERIOR PRODUCTS 36 2.4.7 FORMS WITH COEFFICIENTS IN A VECTOR BUNDLE 36
2.5 LOCAL SOLUTIONS OF D- AND 9-EQUATIONS 38 2.5.1 POINCARE LEMMA 38
2.5.2 DOLBEAULT LEMMA 39 2.5.3 POINCARE LEMMA FOR HOLOMORPHIC FORMS 39
SHEAVES AND COHOMOLOGY 41 3.1 SHEAVES 41 3.2 THE COHOMOLOGY OF SHEAVES
45 3.2.1 THE CANONICAL FLABBY SHEAF CT ASSOCIATED TO A GIVEN SHEAF T 46
3.2.2 RESOLUTIONS OF SHEAVES 47 3.2.3 COHOMOLOGY OF SHEAVES 47 3.3 THE
COHOMOLOGY SEQUENCE ASSOCIATED TO A CLOSED SUBSPACE . . 51 3.4 SOFT AND
FINE SHEAVES 53 3.5 DIRECT IMAGES OF SHEAVES 54 3.6 C-RINGED SPACES 55
3.7 COHERENT SHEAVES 58 HARMONIC FORMS ON HERMITIAN MANIFOLDS 63 4.1
INTRODUCTION 63 4.2 HERMITIAN METRICS ON AN EXTERIOR ALGEBRA 64 4.2.1
HERMITIAN FORMS ON A COMPLEX VECTOR SPACE 64 4.2.2 THE EXTERIOR ALGEBRA
OF V* 66 4.2.3 VOLUME FORM 66 4.2.4 METRICS ON A P Q 67 4.2.5 THE
*-OPERATOR 68 4.2.6 DETERMINATION OF * IN AN ORTHONORMAL BASIS 69 4.3
HERMITIAN METRICS ON A COMPLEX MANIFOLD 70 4.3.1 APPLICATION OF THE
RESULTS OF SECTION 4.2 71 4.4 ADJOINTS OF D, D, D. DE RHAM-HODGE
LAPLACIAN 72 4.4.1 DE RHAM-HODGE OPERATORS 74 4.5 HERMITIAN METRICS AND
LAPLACIAN FOR HOLOMORPHIC BUNDLES . . 75 4.5.1 METRICS ON FORMS WITH
COEFFICIENTS IN A BUNDLE 76 4.5.2 THE ADJOINT OF 5 76 4.5.3 DE
RHAM-HODGE LAPLACE OPERATOR FOR HOLOMORPHIC BUN- DLES 77 4.6 HARMONIC
FORMS AND COHOMOLOGY 78 4.6.1 HARMONIC FORMS ON COMPACT HERMITIAN
MANIFOLD ... 78 4.6.2 THE CASE OF HOLOMORPHIC BUNDLES 79 XV 4.7 DUALITY
81 4.7.1 POINCARE DUALITY 81 4.7.2 SERRE DUALITY 81 4.7.3 APPLICATION TO
MODIFICATIONS ~ 83 HODGE THEORY ON COMPACT KAHLERIAN MANIFOLDS 85 5.1
INTRODUCTION 85 5.2 KAHLERIAN MANIFOLDS 86 5.2.1 LOCAL EXACTNESS 86
5.2.2 COHOMOLOGICAL PROPERTIES OF OJ 88 5.3 LOCAL KAHLERIAN GEOMETRY 89
5.3.1 COVARIANT DERIVATIVES 89 5.3.2 COVARIANT DERIVATIVES OF
DIFFERENTIAL FORMS 93 5.3.3 THE OPERATOR A 94 5.3.4 THE EQUALITY OF THE
DE RHAM-HODGE LAPLACIANS . . . . 95 5.4 THE HODGE DECOMPOSITION ON
COMPACT KAHLERIAN MANIFOLDS . 96 5.4.1 HARMONIC FORMS ON COMPACT
KAHLERIAN MANIFOLDS ... 96 5.5 THE PURE HODGE STRUCTURE ON COHOMOLOGY 97
5.5.1 STRICTNESS FOR D 98 5.5.2 THE CASE OF CLOSED FORMS OF PURE TYPE
100 THE THEORY OF RESIDUES ON A SMOOTH DIVISOR 103 6.1 INTRODUCTION 103
6.2 FORMS WITH LOGARITHMIC SINGULARITIES 103 6.3 THE LONG EXACT HOMOLOGY
RESIDUE SEQUENCE 105 6.3.1 THE LONG EXACT HOMOLOGY SEQUENCE 106 6.3.2
THE RESIDUE FORMULA 106 6.4 THE RESIDUE SEQUENCE IN COHOMOLOGY AND THE
GYSIN MORPHISM 107 6.4.1 DEFINITION OF THE SEQUENCE 107 6.4.2
CONSTRUCTION OF THE GYSIN MORPHISM 108 COMPLEX SPACES 111 7.1 COMPLEX
ANALYTIC VARIETIES AND COMPLEX SPACES ILL 7.2 COHERENT SHEAVES ON
COMPLEX SPACES 114 7.3 MODIFICATIONS AND BLOWING-UP 115 7.4 ALGEBRAIC
AND PROJECTIVE VARIETIES, MOISHEZON SPACES 118 7.5 (B)-KAHLER SPACES 122
7.6 SEMIANALYTIC AND SUBANALYTIC SETS 124 7.7 THE BOREL-MOORE HOMOLOGY
OF A COMPLEX SPACE 127 7.8 SUBANALYTIC CHAINS 129 7.9 INTEGRATION OF
FORMS ON COMPLEX SUBANALYTIC CHAINS 132 7.10 THE MAYER-VIETORIS SEQUENCE
FOR MODIFICATIONS 133 XVI II DIFFERENTIAL FORMS ON COMPLEX SPACES 135 1
THE BASIC EXAMPLE 137 1.1 INTRODUCTION 137 1.2 A RESOLUTION OF C X 138
1.3 THE WEIGHT FILTRATION W 142 1.4 THE SPECTRAL SEQUENCE OF THE
FILTRATION W 143 1.5 THE FILTRATIONS FP AND F 145 1.6 MIXED HODGE
STRUCTURES ON THE COHOMOLOGY AND ON THE SPECTRAL SEQUENCE 146 1.7 CHAINS
AND HOMOLOGY 148 1.8 INTEGRATION OF FORMS ON CHAINS 149 2 DIFFERENTIAL
FORMS ON COMPLEX SPACES 151 2.1 INTRODUCTION 151 2.2 DEFINITIONS AND
STATEMENTS 153 2.2.1 DEFINITION OF THE FAMILY K(X) 153 2.2.2
CONSTRUCTION-EXISTENCE THEOREM 155 2.2.3 DEFINITION OF A PRIMARY
PULLBACK FOR IRREDUCIBLE SPACES 156 2.2.4 DEFINITION OF A PULLBACK
MORPHISM: THE GENERAL CASE . 157 2.2.5 EXISTENCE OF PRIMARY PULLBACK
(THE IRREDUCIBLE CASE) . 160 2.2.6 UNIQUENESS OF PRIMARY PULLBACK (THE
IRREDUCIBLE CASE) 161 2.2.7 EXISTENCE OF PULLBACK: THE GENERAL CASE 161
2.2.8 UNIQUENESS OF PULLBACK: THE GENERAL CASE 161 2.2.9 COMPOSITION OF
PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 161 2.2.10 COMPOSITION OF
PULLBACK: THE GENERAL CASE 161 2.2.11 THE FILTRATION PROPERTY 161 2.3
THE INDUCTION PROCEDURE 162 2.4 THE PROOFS 162 2.4.1 PROOF OF THEOREM
2.7: COMPOSITION OF PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 164 2.4.2
PROOF OF THEOREM 2.8: COMPOSITION OF PULLBACK (THE GEN- ERAL CASE) 165
2.4.3 PROOF OF THEOREM 2.1: CONSTRUCTION-EXISTENCE 167 2.4.4 PROOF OF
THEOREM 2.2: EXISTENCE OF PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 169
2.4.5 PROOF OF THEOREM 2.4: UNIQUENESS OF THE PRIMARY PULL- BACK (THE
IRREDUCIBLE CASE) 172 2.4.6 PROOF OF THEOREM 2.5: EXISTENCE OF PULLBACK
(THE GENERAL CASE) 174 2.4.7 PROOF OF THEOREM 2.6: UNIQUENESS OF THE
PULLBACK (THE GENERAL CASE) 176 2.4.8 PROOF OF THEOREM 2.9: FILTERING
177 2.5 KAHLER HYPERCOVERINGS 177 2.6 CHAINS AND HOMOLOGY 178 XVII 2.7
INTEGRATION OF FORMS ON CHAINS 180 2.8 THE COMPLEX OF GRAUERT AND
GROTHENDIECK 181 3 MIXED HODGE STRUCTURES ON COMPACT SPACES 183 3.1
INTRODUCTION 183 3.2 FILTRATION BY THE DEGREE: THE WEIGHT FILTRATION 184
3.3 THE WEIGHT FILTRATION IN COHOMOLOGY 186 3.4 THE ACTION OF D ON THE
FILTERED COMPLEXES 186 3.5 THE FIRST TERM OF THE SPECTRAL SEQUENCE 189
3.6 THE SECOND TERM OF THE SPECTRAL SEQUENCE 190 3.7 COMPUTATION OF D R
192 3.8 THE FILTRATIONS F P AND F 194 3.9 PURE HODGE STRUCTURES ON THE
SPECTRAL SEQUENCE 195 3.10 THE HODGE FILTRATIONS ON E* K 199 3.11 THE
MIXED HODGE STRUCTURE ON THE COHOMOLOGY 202 3.12 THE MAYER-VIETORIS
SEQUENCE 203 3.13 THE DIFFERENTIAL D IS A STRICT MORPHISM FOR THE
FILTRATION F P . 205 III MIXED HODGE STRUCTURES ON NONCOMPACT SPACES 209
1 RESIDUES AND HODGE MIXED STRUCTURES: LERAY THEORY 211 1.1 INTRODUCTION
211 1.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 213 1.2.1 DEFINITION OF
X (LOG D) 213 1.2.2 FILTRATION BY THE ORDER OF POLES 213 1.3 RESIDUES
(CLASSICAL LERAY THEORY) 215 1.3.1 THE RESIDUES IN LOCAL COHOMOLOGY 215
1.3.2 THE RESIDUES IN GLOBAL COHOMOLOGY 217 1.3.3 THE COHOMOLOGY OF X
D 218 1.4 RESIDUES AND MIXED HODGE STRUCTURES (THE CASE OF A SMOOTH
DIVISOR) 219 1.4.1 HODGE FILTRATIONS AND RESIDUES 219 1.4.2 PURE HODGE
STRUCTURE ON E L { K = E L { K {X) 221 1.4.3 MIXED HODGE STRUCTURE ON H
K (X D,C) 224 1.4.4 FUNCTORIALITY * 225 1.4.5 OTHER RESIDUES 226 2
RESIDUES AND MIXED HODGE STRUCTURES ON NONCOMPACT MANI- FOLDS 227 2.1
INTRODUCTION 227 2.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 229 2.2.1
DEFINITION OF X (LOG D) 229 2.2.2 FILTRATION BY THE ORDER OF POLES :
230 2.2.3 THE FILTRATION W ON Q KX (LOG D) 231 XVM 2.2.4 THE DE RHAM
COMPLEX OF A DIVISOR 232 2.3 RESIDUES (SMOOTH CASE) 233 2.3.1 THE
RESIDUES IN LOCAL COHOMOLOGY 235 2.3.2 THE RESIDUES IN GLOBAL COHOMOLOGY
237 2.3.3 THE COHOMOLOGY OF X D 237 2.4 RESIDUES AND MIXED HODGE
STRUCTURES (THE SMOOTH CASE) . . 239 2.4.1 HODGE FILTRATIONS AND
RESIDUES 239 2.4.2 PURE HODGE STRUCTURE ON E L { K (X) 241 2.4.3
DEGENERATION OF THE SPECTRAL SEQUENCE 245 2.5 THE STRICTNESS OF DO AND D
WITH RESPECT TO THE HODGE FILTRATION 249 2.5.1 THE CONJUGATE COMPLEX 249
2.5.2 STRICTNESS OF D 0 251 2.5.3 THE RECURSIVE AND THE DIRECT
FILTRATIONS ON E L2 K (X) . . 253 2.5.4 STRICTNESS OF D 257 MIXED
HODGE STRUCTURES ON NONCOMPACT SPACES: THE BASIC EXAMPLE 259 3.1
INTRODUCTION 259 3.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 260 3.2.1
THE COHOMOLOGY OF X Q 261 3.2.2 FILTRATION W BY THE ORDER OF POLES 263
3.3 RESIDUES (QUASI-SMOOTH CASE) 264 3.4 THE RESIDUE COMPLEX 266 3.5
RESIDUES AND MIXED HODGE STRUCTURES (QUASI-SMOOTH CASE) . 267 3.5.1
HODGE FILTRATIONS AND RESIDUES 267 3.5.2 PURE HODGE STRUCTURE ON E L { K
(X) 269 3.5.3 THE DIFFERENTIAL D X 270 3.5.4 THE CONJUGATE COMPLEX 272
3.5.5 DEGENERATION OF THE SPECTRAL SEQUENCE 273 MIXED HODGE STRUCTURES
ON NONCOMPACT SINGULAR SPACES 277 4.1 INTRODUCTION . 277 4.2 LOGARITHMIC
COMPLEXES ON SINGULAR SPACES 278 4.2.1 LOGARITHMIC FORMS 278 4.3 THE
WEIGHT FILTRATION W 283 4.3.1 ON A LOGARITHMIC DE RHAM COMPLEX WHEN X IS
SMOOTH 283 4.3.2 ON A GENERAL LOGARITHMIC COMPLEX 284 4.3.3 FILTRATION
OF THE COHOMOLOGY AND SPECTRAL SEQUENCE . . 285 4.4 RESIDUES 287 4.4.1
RESIDUES ON THE GRADED COMPLEXES 287 4.4.2 GLOBAL RESIDUES ON THE
FILTERED COMPLEXES 289 4.5 RESIDUES AND MIXED HODGE STRUCTURE (SINGULAR
CASE) .... 290 4.5.1 SHIFTED HODGE FILTRATIONS AND RESIDUES 290 4.5.2
PURE HODGE STRUCTURE ON * FC 293 4.5.3 THE DIFFERENTIAL DI 294 XIX
4.5.4 THE CONJUGATE COMPLEX 296 4.6 DEGENERATION OF THE SPECTRAL
SEQUENCE 297 4.7 STRICTNESS OF D 302 REFERENCES 307
|
| adam_txt |
DIFFERENTIAL FORMS ON SINGULAR VARIETIES DE RHAM AND HODGE THEORY
SIMPLIFIED VINCENZO ANCONA UNIVERSITY OFFIRENZE FLORENCE, ITALY BERNARD
GAVEAU UNIVERSITY PIERRE ET MARIE CURIE PARIS, FRANCE CHAPMAN & HALL/CRC
TAYLOR &. FRANCIS GROUP BOCA RATON LONDON NEW YORK SINGAPORE CONTENTS I
CLASSICAL HODGE THEORY 1 1 SPECTRAL SEQUENCES AND MIXED HODGE STRUCTURES
3 1.1 INTRODUCTION 3 1.2 FILTRATIONS 3 1.3 STRICT MORPHISMS 4 1.4
FILTERED COMPLEXES 5 1.5 SPECTRAL SEQUENCES 5 1.6 THE FIRST TERM OF THE
SPECTRAL SEQUENCE 6 1.7 THE GRADED COHOMOLOGY 7 1.8 PURE HODGE
STRUCTURES 8 1.9 MORPHISMS OF PURE HODGE STRUCTURES 9 1.10 MIXED HODGE
STRUCTURES 11 1.11 EXACT SEQUENCES OF MIXED HODGE STRUCTURES 14 1.12
SHIFTED COMPLEXES AND SHIFTED FILTRATIONS 15 1.13 THE STRICTNESS OF D
AND THE DEGENERATION OF THE SPECTRAL SE- QUENCE 15 1.14 FLAT MODULES 16
1.15 CONNECTING HOMOMORPHISMS 17 2 COMPLEX MANIFOLDS, VECTOR BUNDLES,
DIFFERENTIAL FORMS 19 2.1 INTRODUCTION 19 2.2 COMPLEX MANIFOLDS 19 2.2.1
DEFINITIONS OF COMPLEX COORDINATES AND MANIFOLDS . 19 2.2.2 TANGENT
VECTORS 20 2.2.3 HOLOMORPHIC FUNCTIONS . 22 2.2.4 COMPLEX SUBMANIFOLDS
22 2.2.5 EXAMPLES 23 2.3 COMPLEX VECTOR BUNDLES AND DIVISORS 25 2.3.1
OPERATIONS ON BUNDLES . 27 2.3.2 TANGENT BUNDLE 28 2.3.3 EXAMPLE:
COMPLEX TORI 29 2.3.4 LINE BUNDLES AND DIVISORS 29 2.3.5 EXAMPLE: P* AND
ITS LINE BUNDLES 30 2.4 DIFFERENTIAL FORMS ON COMPLEX MANIFOLDS . . . 31
2.4.1 EXPRESSIONS IN LOCAL COORDINATES 32 XM XIV 2.4.2 THE HODGE
FILTRATIONS F AND F 33 2.4.3 PULLBACK 33 2.4.4 EXTERIOR DIFFERENTIALS 35
2.4.5 EXTERIOR DIFFERENTIALS AND PULLBACK 36 2.4.6 DIFFERENTIALS AND
EXTERIOR PRODUCTS 36 2.4.7 FORMS WITH COEFFICIENTS IN A VECTOR BUNDLE 36
2.5 LOCAL SOLUTIONS OF D- AND 9-EQUATIONS 38 2.5.1 POINCARE LEMMA 38
2.5.2 DOLBEAULT LEMMA 39 2.5.3 POINCARE LEMMA FOR HOLOMORPHIC FORMS 39
SHEAVES AND COHOMOLOGY 41 3.1 SHEAVES 41 3.2 THE COHOMOLOGY OF SHEAVES
45 3.2.1 THE CANONICAL FLABBY SHEAF CT ASSOCIATED TO A GIVEN SHEAF T 46
3.2.2 RESOLUTIONS OF SHEAVES 47 3.2.3 COHOMOLOGY OF SHEAVES 47 3.3 THE
COHOMOLOGY SEQUENCE ASSOCIATED TO A CLOSED SUBSPACE . . 51 3.4 SOFT AND
FINE SHEAVES 53 3.5 DIRECT IMAGES OF SHEAVES 54 3.6 C-RINGED SPACES 55
3.7 COHERENT SHEAVES 58 HARMONIC FORMS ON HERMITIAN MANIFOLDS 63 4.1
INTRODUCTION 63 4.2 HERMITIAN METRICS ON AN EXTERIOR ALGEBRA 64 4.2.1
HERMITIAN FORMS ON A COMPLEX VECTOR SPACE 64 4.2.2 THE EXTERIOR ALGEBRA
OF V* "\66 4.2.3 VOLUME FORM 66 4.2.4 METRICS ON A P Q 67 4.2.5 THE
*-OPERATOR 68 4.2.6 DETERMINATION OF * IN AN ORTHONORMAL BASIS 69 4.3
HERMITIAN METRICS ON A COMPLEX MANIFOLD 70 4.3.1 APPLICATION OF THE
RESULTS OF SECTION 4.2 71 4.4 ADJOINTS OF D, D, D. DE RHAM-HODGE
LAPLACIAN 72 4.4.1 DE RHAM-HODGE OPERATORS 74 4.5 HERMITIAN METRICS AND
LAPLACIAN FOR HOLOMORPHIC BUNDLES . . 75 4.5.1 METRICS ON FORMS WITH
COEFFICIENTS IN A BUNDLE 76 4.5.2 THE ADJOINT OF 5 76 4.5.3 DE
RHAM-HODGE LAPLACE OPERATOR FOR HOLOMORPHIC BUN- DLES 77 4.6 HARMONIC
FORMS AND COHOMOLOGY 78 4.6.1 HARMONIC FORMS ON COMPACT HERMITIAN
MANIFOLD . 78 4.6.2 THE CASE OF HOLOMORPHIC BUNDLES 79 XV 4.7 DUALITY
81 4.7.1 POINCARE DUALITY 81 4.7.2 SERRE DUALITY 81 4.7.3 APPLICATION TO
MODIFICATIONS ~ " 83 HODGE THEORY ON COMPACT KAHLERIAN MANIFOLDS 85 5.1
INTRODUCTION 85 5.2 KAHLERIAN MANIFOLDS 86 5.2.1 LOCAL EXACTNESS 86
5.2.2 COHOMOLOGICAL PROPERTIES OF OJ 88 5.3 LOCAL KAHLERIAN GEOMETRY 89
5.3.1 COVARIANT DERIVATIVES 89 5.3.2 COVARIANT DERIVATIVES OF
DIFFERENTIAL FORMS 93 5.3.3 THE OPERATOR A 94 5.3.4 THE EQUALITY OF THE
DE RHAM-HODGE LAPLACIANS . . . . 95 5.4 THE HODGE DECOMPOSITION ON
COMPACT KAHLERIAN MANIFOLDS . 96 5.4.1 HARMONIC FORMS ON COMPACT
KAHLERIAN MANIFOLDS . 96 5.5 THE PURE HODGE STRUCTURE ON COHOMOLOGY 97
5.5.1 STRICTNESS FOR D 98 5.5.2 THE CASE OF CLOSED FORMS OF PURE TYPE
100 THE THEORY OF RESIDUES ON A SMOOTH DIVISOR 103 6.1 INTRODUCTION 103
6.2 FORMS WITH LOGARITHMIC SINGULARITIES 103 6.3 THE LONG EXACT HOMOLOGY
RESIDUE SEQUENCE 105 6.3.1 THE LONG EXACT HOMOLOGY SEQUENCE 106 6.3.2
THE RESIDUE FORMULA 106 6.4 THE RESIDUE SEQUENCE IN COHOMOLOGY AND THE
GYSIN MORPHISM 107 6.4.1 DEFINITION OF THE SEQUENCE 107 6.4.2
CONSTRUCTION OF THE GYSIN MORPHISM 108 COMPLEX SPACES 111 7.1 COMPLEX
ANALYTIC VARIETIES AND COMPLEX SPACES ILL 7.2 COHERENT SHEAVES ON
COMPLEX SPACES 114 7.3 MODIFICATIONS AND BLOWING-UP 115 7.4 ALGEBRAIC
AND PROJECTIVE VARIETIES, MOISHEZON SPACES 118 7.5 (B)-KAHLER SPACES 122
7.6 SEMIANALYTIC AND SUBANALYTIC SETS 124 7.7 THE BOREL-MOORE HOMOLOGY
OF A COMPLEX SPACE 127 7.8 SUBANALYTIC CHAINS 129 7.9 INTEGRATION OF
FORMS ON COMPLEX SUBANALYTIC CHAINS 132 7.10 THE MAYER-VIETORIS SEQUENCE
FOR MODIFICATIONS 133 XVI II DIFFERENTIAL FORMS ON COMPLEX SPACES 135 1
THE BASIC EXAMPLE 137 1.1 INTRODUCTION 137 1.2 A RESOLUTION OF C X 138
1.3 THE WEIGHT FILTRATION W 142 1.4 THE SPECTRAL SEQUENCE OF THE
FILTRATION W 143 1.5 THE FILTRATIONS FP AND F" 145 1.6 MIXED HODGE
STRUCTURES ON THE COHOMOLOGY AND ON THE SPECTRAL SEQUENCE 146 1.7 CHAINS
AND HOMOLOGY 148 1.8 INTEGRATION OF FORMS ON CHAINS 149 2 DIFFERENTIAL
FORMS ON COMPLEX SPACES 151 2.1 INTRODUCTION 151 2.2 DEFINITIONS AND
STATEMENTS 153 2.2.1 DEFINITION OF THE FAMILY K(X) 153 2.2.2
CONSTRUCTION-EXISTENCE THEOREM 155 2.2.3 DEFINITION OF A PRIMARY
PULLBACK FOR IRREDUCIBLE SPACES 156 2.2.4 DEFINITION OF A PULLBACK
MORPHISM: THE GENERAL CASE . 157 2.2.5 EXISTENCE OF PRIMARY PULLBACK
(THE IRREDUCIBLE CASE) . 160 2.2.6 UNIQUENESS OF PRIMARY PULLBACK (THE
IRREDUCIBLE CASE) 161 2.2.7 EXISTENCE OF PULLBACK: THE GENERAL CASE 161
2.2.8 UNIQUENESS OF PULLBACK: THE GENERAL CASE 161 2.2.9 COMPOSITION OF
PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 161 2.2.10 COMPOSITION OF
PULLBACK: THE GENERAL CASE 161 2.2.11 THE FILTRATION PROPERTY 161 2.3
THE INDUCTION PROCEDURE 162 2.4 THE PROOFS 162 2.4.1 PROOF OF THEOREM
2.7: COMPOSITION OF PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 164 2.4.2
PROOF OF THEOREM 2.8: COMPOSITION OF PULLBACK (THE GEN- ERAL CASE) 165
2.4.3 PROOF OF THEOREM 2.1: CONSTRUCTION-EXISTENCE 167 2.4.4 PROOF OF
THEOREM 2.2: EXISTENCE OF PRIMARY PULLBACK (THE IRREDUCIBLE CASE) 169
2.4.5 PROOF OF THEOREM 2.4: UNIQUENESS OF THE PRIMARY PULL- BACK (THE
IRREDUCIBLE CASE) 172 2.4.6 PROOF OF THEOREM 2.5: EXISTENCE OF PULLBACK
(THE GENERAL CASE) 174 2.4.7 PROOF OF THEOREM 2.6: UNIQUENESS OF THE
PULLBACK (THE GENERAL CASE) 176 2.4.8 PROOF OF THEOREM 2.9: FILTERING
177 2.5 KAHLER HYPERCOVERINGS 177 2.6 CHAINS AND HOMOLOGY 178 XVII 2.7
INTEGRATION OF FORMS ON CHAINS 180 2.8 THE COMPLEX OF GRAUERT AND
GROTHENDIECK 181 3 MIXED HODGE STRUCTURES ON COMPACT SPACES 183 3.1
INTRODUCTION 183 3.2 FILTRATION BY THE DEGREE: THE WEIGHT FILTRATION 184
3.3 THE WEIGHT FILTRATION IN COHOMOLOGY 186 3.4 THE ACTION OF D ON THE
FILTERED COMPLEXES 186 3.5 THE FIRST TERM OF THE SPECTRAL SEQUENCE 189
3.6 THE SECOND TERM OF THE SPECTRAL SEQUENCE 190 3.7 COMPUTATION OF D R
192 3.8 THE FILTRATIONS F P AND F" 194 3.9 PURE HODGE STRUCTURES ON THE
SPECTRAL SEQUENCE 195 3.10 THE HODGE FILTRATIONS ON E* K 199 3.11 THE
MIXED HODGE STRUCTURE ON THE COHOMOLOGY 202 3.12 THE MAYER-VIETORIS
SEQUENCE 203 3.13 THE DIFFERENTIAL D IS A STRICT MORPHISM FOR THE
FILTRATION F P . 205 III MIXED HODGE STRUCTURES ON NONCOMPACT SPACES 209
1 RESIDUES AND HODGE MIXED STRUCTURES: LERAY THEORY 211 1.1 INTRODUCTION
211 1.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 213 1.2.1 DEFINITION OF
X (LOG D) 213 1.2.2 FILTRATION BY THE ORDER OF POLES 213 1.3 RESIDUES
(CLASSICAL LERAY THEORY) 215 1.3.1 THE RESIDUES IN LOCAL COHOMOLOGY 215
1.3.2 THE RESIDUES IN GLOBAL COHOMOLOGY 217 1.3.3 THE COHOMOLOGY OF X \
D 218 1.4 RESIDUES AND MIXED HODGE STRUCTURES (THE CASE OF A SMOOTH
DIVISOR) 219 1.4.1 HODGE FILTRATIONS AND RESIDUES 219 1.4.2 PURE HODGE
STRUCTURE ON E L { K = E L { K {X) 221 1.4.3 MIXED HODGE STRUCTURE ON H
K (X\D,C) 224 1.4.4 FUNCTORIALITY * 225 1.4.5 OTHER RESIDUES 226 2
RESIDUES AND MIXED HODGE STRUCTURES ON NONCOMPACT MANI- FOLDS 227 2.1
INTRODUCTION 227 2.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 229 2.2.1
DEFINITION OF X (LOG D) 229 2.2.2 FILTRATION BY THE ORDER OF POLES :
230 2.2.3 THE FILTRATION W ON Q KX (LOG D) 231 XVM 2.2.4 THE DE RHAM
COMPLEX OF A DIVISOR 232 2.3 RESIDUES (SMOOTH CASE) 233 2.3.1 THE
RESIDUES IN LOCAL COHOMOLOGY 235 2.3.2 THE RESIDUES IN GLOBAL COHOMOLOGY
237 2.3.3 THE COHOMOLOGY OF X \ D 237 2.4 RESIDUES AND MIXED HODGE
STRUCTURES (THE SMOOTH CASE) . . 239 2.4.1 HODGE FILTRATIONS AND
RESIDUES 239 2.4.2 PURE HODGE STRUCTURE ON E L { K (X) 241 2.4.3
DEGENERATION OF THE SPECTRAL SEQUENCE 245 2.5 THE STRICTNESS OF DO AND D
WITH RESPECT TO THE HODGE FILTRATION 249 2.5.1 THE CONJUGATE COMPLEX 249
2.5.2 STRICTNESS OF D 0 251 2.5.3 THE RECURSIVE AND THE DIRECT
FILTRATIONS ON E L2 ' K (X) . . 253 2.5.4 STRICTNESS OF D 257 MIXED
HODGE STRUCTURES ON NONCOMPACT SPACES: THE BASIC EXAMPLE 259 3.1
INTRODUCTION 259 3.2 THE STANDARD LOGARITHMIC DE RHAM COMPLEX 260 3.2.1
THE COHOMOLOGY OF X \ Q 261 3.2.2 FILTRATION W BY THE ORDER OF POLES 263
3.3 RESIDUES (QUASI-SMOOTH CASE) 264 3.4 THE RESIDUE COMPLEX 266 3.5
RESIDUES AND MIXED HODGE STRUCTURES (QUASI-SMOOTH CASE) . 267 3.5.1
HODGE FILTRATIONS AND RESIDUES 267 3.5.2 PURE HODGE STRUCTURE ON E L { K
(X) 269 3.5.3 THE DIFFERENTIAL D X 270 3.5.4 THE CONJUGATE COMPLEX 272
3.5.5 DEGENERATION OF THE SPECTRAL SEQUENCE 273 MIXED HODGE STRUCTURES
ON NONCOMPACT SINGULAR SPACES 277 4.1 INTRODUCTION . 277 4.2 LOGARITHMIC
COMPLEXES ON SINGULAR SPACES 278 4.2.1 LOGARITHMIC FORMS 278 4.3 THE
WEIGHT FILTRATION W 283 4.3.1 ON A LOGARITHMIC DE RHAM COMPLEX WHEN X IS
SMOOTH 283 4.3.2 ON A GENERAL LOGARITHMIC COMPLEX 284 4.3.3 FILTRATION
OF THE COHOMOLOGY AND SPECTRAL SEQUENCE . . 285 4.4 RESIDUES 287 4.4.1
RESIDUES ON THE GRADED COMPLEXES 287 4.4.2 GLOBAL RESIDUES ON THE
FILTERED COMPLEXES 289 4.5 RESIDUES AND MIXED HODGE STRUCTURE (SINGULAR
CASE) . 290 4.5.1 SHIFTED HODGE FILTRATIONS AND RESIDUES 290 4.5.2
PURE HODGE STRUCTURE ON *' FC 293 4.5.3 THE DIFFERENTIAL DI 294 XIX
4.5.4 THE CONJUGATE COMPLEX 296 4.6 DEGENERATION OF THE SPECTRAL
SEQUENCE 297 4.7 STRICTNESS OF D 302 REFERENCES 307 |
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| callnumber-label | QA381 |
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| classification_rvk | SK 780 |
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| dewey-full | 515/.37 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.37 |
| dewey-search | 515/.37 |
| dewey-sort | 3515 237 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV021506335 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:17:15Z |
| indexdate | 2024-07-09T20:37:21Z |
| institution | BVB |
| isbn | 0849337399 9780849337390 |
| language | English |
| lccn | 2005048467 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014723006 |
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| physical | xix, 311 p. graph. Darst. 24 cm |
| publishDate | 2006 |
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| publishDateSort | 2006 |
| publisher | Chapman & Hall/CRC |
| record_format | marc |
| series | Monographs and textbooks in pure and applied mathematics |
| series2 | Monographs and textbooks in pure and applied mathematics |
| spelling | Ancona, Vincenzo Verfasser aut Differential forms on singular varieties De Rham and Hodge theory simplified Vincenzo Ancona ; Bernard Gaveau Boca Raton, FL [u.a.] Chapman & Hall/CRC 2006 xix, 311 p. graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Monographs and textbooks in pure and applied mathematics 273 Includes bibliographical references (p. 307-[310]) and index Formes différentielles Géométrie algébrique Hodge, Théorie de Singularités (Mathématiques) Differential forms Hodge theory Singularities (Mathematics) Geometry, Algebraic Differentialform (DE-588)4149772-7 gnd rswk-swf Hodge-Theorie (DE-588)4135967-7 gnd rswk-swf Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd rswk-swf DeRham-Kohomologie (DE-588)4352640-8 gnd rswk-swf Differentialform (DE-588)4149772-7 s Hodge-Theorie (DE-588)4135967-7 s DeRham-Kohomologie (DE-588)4352640-8 s Kähler-Mannigfaltigkeit (DE-588)4162978-4 s b DE-604 Gaveau, Bernard Verfasser aut Monographs and textbooks in pure and applied mathematics 273 (DE-604)BV000001885 273 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014723006&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ancona, Vincenzo Gaveau, Bernard Differential forms on singular varieties De Rham and Hodge theory simplified Monographs and textbooks in pure and applied mathematics Formes différentielles Géométrie algébrique Hodge, Théorie de Singularités (Mathématiques) Differential forms Hodge theory Singularities (Mathematics) Geometry, Algebraic Differentialform (DE-588)4149772-7 gnd Hodge-Theorie (DE-588)4135967-7 gnd Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd DeRham-Kohomologie (DE-588)4352640-8 gnd |
| subject_GND | (DE-588)4149772-7 (DE-588)4135967-7 (DE-588)4162978-4 (DE-588)4352640-8 |
| title | Differential forms on singular varieties De Rham and Hodge theory simplified |
| title_auth | Differential forms on singular varieties De Rham and Hodge theory simplified |
| title_exact_search | Differential forms on singular varieties De Rham and Hodge theory simplified |
| title_exact_search_txtP | Differential forms on singular varieties De Rham and Hodge theory simplified |
| title_full | Differential forms on singular varieties De Rham and Hodge theory simplified Vincenzo Ancona ; Bernard Gaveau |
| title_fullStr | Differential forms on singular varieties De Rham and Hodge theory simplified Vincenzo Ancona ; Bernard Gaveau |
| title_full_unstemmed | Differential forms on singular varieties De Rham and Hodge theory simplified Vincenzo Ancona ; Bernard Gaveau |
| title_short | Differential forms on singular varieties |
| title_sort | differential forms on singular varieties de rham and hodge theory simplified |
| title_sub | De Rham and Hodge theory simplified |
| topic | Formes différentielles Géométrie algébrique Hodge, Théorie de Singularités (Mathématiques) Differential forms Hodge theory Singularities (Mathematics) Geometry, Algebraic Differentialform (DE-588)4149772-7 gnd Hodge-Theorie (DE-588)4135967-7 gnd Kähler-Mannigfaltigkeit (DE-588)4162978-4 gnd DeRham-Kohomologie (DE-588)4352640-8 gnd |
| topic_facet | Formes différentielles Géométrie algébrique Hodge, Théorie de Singularités (Mathématiques) Differential forms Hodge theory Singularities (Mathematics) Geometry, Algebraic Differentialform Hodge-Theorie Kähler-Mannigfaltigkeit DeRham-Kohomologie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014723006&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001885 |
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