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The Resource Hodge Theory (MN-49)

Hodge Theory (MN-49)

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Summary
This book provides a comprehensive and up-to-date introduction to Hodge theory-one of the central and most vibrant areas of contemporary mathematics-from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students
Language
eng
Work
Publication
Extent
1 online resource (608 pages)
Note
3.1.6 Cohomology Class of a Subvariety and Hodge Conjecture
Contents
  • Cover; Title; Copyright; Contributors; Contributors; Contents; Preface; 1 Kähler Manifolds; 1.1 Complex Manifolds; 1.1.1 Definition and Examples; 1.1.2 Holomorphic Vector Bundles; 1.2 Differential Forms on Complex Manifolds; 1.2.1 Almost Complex Manifolds; 1.2.2 Tangent and Cotangent Space; 1.2.3 De Rham and Dolbeault Cohomologies; 1.3 Symplectic, Hermitian, and Kähler Structures; 1.3.1 Kähler Manifolds; 1.3.2 The Chern Class of a Holomorphic Line Bundle; 1.4 Harmonic Forms-Hodge Theorem; 1.4.1 Compact Real Manifolds; 1.4.2 The [del symbol] -Laplacian; 1.5 Cohomology of Compact Kähler Manifolds
  • 1.5.1 The Kähler Identities1.5.2 The Hodge Decomposition Theorem; 1.5.3 Lefschetz Theorems and Hodge-Riemann Bilinear Relations; A Linear Algebra; A.1 Real and Complex Vector Spaces; A.2 The Weight Filtration of a Nilpotent Transformation; A.3 Representations of sl(2,C) and Lefschetz Theorems; A.4 Hodge Structures; B The Kähler Identities; B.1 Symplectic Linear Algebra; B.2 Compatible Inner Products; B.3 Symplectic Manifolds; B.4 The Kähler Identities; Bibliography; 2 The Algebraic de Rham Theorem; Introduction; Part I. Sheaf Cohomology, Hypercohomology, and the Projective Case; 2.1 Sheaves
  • 2.1.1 The Étalé Space of a Presheaf2.1.2 Exact Sequences of Sheaves; 2.1.3 Resolutions; 2.2 Sheaf Cohomology; 2.2.1 Godement's Canonical Resolution; 2.2.2 Cohomology with Coefficients in a Sheaf; 2.2.3 Flasque Sheaves; 2.2.4 Cohomology Sheaves and Exact Functors; 2.2.5 Fine Sheaves; 2.2.6 Cohomology with Coefficients in a Fine Sheaf; 2.3 Coherent Sheaves and Serre's GAGA Principle; 2.4 The Hypercohomology of a Complex of Sheaves; 2.4.1 The Spectral Sequences of Hypercohomology; 2.4.2 Acyclic Resolutions; 2.5 The Analytic de Rham Theorem; 2.5.1 The Holomorphic Poincaré Lemma
  • 2.10 The Algebraic de Rham Theorem for an Affine Variety2.10.1 The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms; 2.10.2 The Hypercohomology of Rational and Meromorphic Forms; 2.10.3 Comparison of Meromorphic and Smooth Forms; Bibliography; 3 Mixed Hodge Structures; 3.1 Hodge Structure on a Smooth Compact Complex Variety; 3.1.1 Hodge Structure (HS); 3.1.2 Spectral Sequence of a Filtered Complex; 3.1.3 Hodge Structure on the Cohomology of Nonsingular Compact Complex Algebraic Varieties; 3.1.4 Lefschetz Decomposition and Polarized Hodge Structure; 3.1.5 Examples
  • 2.5.2 The Analytic de Rham Theorem2.6 The Algebraic de Rham Theorem for a Projective Variety; Part II. Čech Cohomology and the Algebraic de Rham Theorem in General; 2.7 Čech Cohomology of a Sheaf; 2.7.1 Čech Cohomology of an Open Cover; 2.7.2 Relation Between Čech Cohomology and Sheaf Cohomology; 2.8 Čech Cohomology of a Complex of Sheaves; 2.8.1 The Relation Between Čech Cohomology and Hypercohomology; 2.9 Reduction to the Affine Case; 2.9.1 Proof that the General Case Implies the Affine Case; 2.9.2 Proof that the Affine Case Implies the General Case
Isbn
9781306783668
Instance
Label
Hodge Theory (MN-49)
Title
Hodge Theory (MN-49)
Contributor
Subject
Genre
Language
eng
Summary
This book provides a comprehensive and up-to-date introduction to Hodge theory-one of the central and most vibrant areas of contemporary mathematics-from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students
Member of
Cataloging source
EBLCP
Dewey number
516.36
Index
no index present
LC call number
  • QA3
  • QA613
Literary form
non fiction
Nature of contents
dictionaries
Series statement
Mathematical Notes
Label
Hodge Theory (MN-49)
Publication
Note
3.1.6 Cohomology Class of a Subvariety and Hodge Conjecture
Antecedent source
unknown
http://library.link/vocab/branchCode
  • net
Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Cover; Title; Copyright; Contributors; Contributors; Contents; Preface; 1 Kähler Manifolds; 1.1 Complex Manifolds; 1.1.1 Definition and Examples; 1.1.2 Holomorphic Vector Bundles; 1.2 Differential Forms on Complex Manifolds; 1.2.1 Almost Complex Manifolds; 1.2.2 Tangent and Cotangent Space; 1.2.3 De Rham and Dolbeault Cohomologies; 1.3 Symplectic, Hermitian, and Kähler Structures; 1.3.1 Kähler Manifolds; 1.3.2 The Chern Class of a Holomorphic Line Bundle; 1.4 Harmonic Forms-Hodge Theorem; 1.4.1 Compact Real Manifolds; 1.4.2 The [del symbol] -Laplacian; 1.5 Cohomology of Compact Kähler Manifolds
  • 1.5.1 The Kähler Identities1.5.2 The Hodge Decomposition Theorem; 1.5.3 Lefschetz Theorems and Hodge-Riemann Bilinear Relations; A Linear Algebra; A.1 Real and Complex Vector Spaces; A.2 The Weight Filtration of a Nilpotent Transformation; A.3 Representations of sl(2,C) and Lefschetz Theorems; A.4 Hodge Structures; B The Kähler Identities; B.1 Symplectic Linear Algebra; B.2 Compatible Inner Products; B.3 Symplectic Manifolds; B.4 The Kähler Identities; Bibliography; 2 The Algebraic de Rham Theorem; Introduction; Part I. Sheaf Cohomology, Hypercohomology, and the Projective Case; 2.1 Sheaves
  • 2.1.1 The Étalé Space of a Presheaf2.1.2 Exact Sequences of Sheaves; 2.1.3 Resolutions; 2.2 Sheaf Cohomology; 2.2.1 Godement's Canonical Resolution; 2.2.2 Cohomology with Coefficients in a Sheaf; 2.2.3 Flasque Sheaves; 2.2.4 Cohomology Sheaves and Exact Functors; 2.2.5 Fine Sheaves; 2.2.6 Cohomology with Coefficients in a Fine Sheaf; 2.3 Coherent Sheaves and Serre's GAGA Principle; 2.4 The Hypercohomology of a Complex of Sheaves; 2.4.1 The Spectral Sequences of Hypercohomology; 2.4.2 Acyclic Resolutions; 2.5 The Analytic de Rham Theorem; 2.5.1 The Holomorphic Poincaré Lemma
  • 2.10 The Algebraic de Rham Theorem for an Affine Variety2.10.1 The Hypercohomology of the Direct Image of a Sheaf of Smooth Forms; 2.10.2 The Hypercohomology of Rational and Meromorphic Forms; 2.10.3 Comparison of Meromorphic and Smooth Forms; Bibliography; 3 Mixed Hodge Structures; 3.1 Hodge Structure on a Smooth Compact Complex Variety; 3.1.1 Hodge Structure (HS); 3.1.2 Spectral Sequence of a Filtered Complex; 3.1.3 Hodge Structure on the Cohomology of Nonsingular Compact Complex Algebraic Varieties; 3.1.4 Lefschetz Decomposition and Polarized Hodge Structure; 3.1.5 Examples
  • 2.5.2 The Analytic de Rham Theorem2.6 The Algebraic de Rham Theorem for a Projective Variety; Part II. Čech Cohomology and the Algebraic de Rham Theorem in General; 2.7 Čech Cohomology of a Sheaf; 2.7.1 Čech Cohomology of an Open Cover; 2.7.2 Relation Between Čech Cohomology and Sheaf Cohomology; 2.8 Čech Cohomology of a Complex of Sheaves; 2.8.1 The Relation Between Čech Cohomology and Hypercohomology; 2.9 Reduction to the Affine Case; 2.9.1 Proof that the General Case Implies the Affine Case; 2.9.2 Proof that the Affine Case Implies the General Case
Control code
000054658862
Dimensions
unknown
Extent
1 online resource (608 pages)
File format
unknown
Form of item
online
Isbn
9781306783668
Level of compression
unknown
Media category
computer
Media MARC source
rdamedia
Media type code
c
http://library.link/vocab/ext/overdrive/overdriveId
22573/ctt5xt5cc
Quality assurance targets
not applicable
http://library.link/vocab/recordID
.b32769908
Reformatting quality
unknown
Sound
unknown sound
Specific material designation
remote
System control number
  • (OCoLC)880057790
  • ebsco1400851475
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