The Resource Algebraic Surfaces and Holomorphic Vector Bundles, by Robert Friedman, (electronic resource)

Algebraic Surfaces and Holomorphic Vector Bundles, by Robert Friedman, (electronic resource)

Label
Algebraic Surfaces and Holomorphic Vector Bundles
Title
Algebraic Surfaces and Holomorphic Vector Bundles
Statement of responsibility
by Robert Friedman
Creator
Author
Author
Subject
Language
  • eng
  • eng
Summary
This book is based on courses given at Columbia University on vector bun­ dles (1988) and on the theory of algebraic surfaces (1992), as well as lectures in the Park City lIAS Mathematics Institute on 4-manifolds and Donald­ son invariants. The goal of these lectures was to acquaint researchers in 4-manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic bundles on algebraic surfaces with a view toward computing Donaldson invariants. Since that time, the focus of 4-manifold topology has shifted dramatically, at first be­ cause topological methods have largely superseded algebro-geometric meth­ ods in computing Donaldson invariants, and more importantly because of and Witten, which have greatly sim­ the new invariants defined by Seiberg plified the theory and led to proofs of the basic conjectures concerning the 4-manifold topology of algebraic surfaces. However, the study of algebraic surfaces and the moduli spaces of bundles on them remains a fundamen­ tal problem in algebraic geometry, and I hope that this book will make this subject more accessible. Moreover, the recent applications of Seiberg­ Witten theory to symplectic 4-manifolds suggest that there is room for yet another treatment of the classification of algebraic surfaces. In particular, despite the number of excellent books concerning algebraic surfaces, I hope that the half of this book devoted to them will serve as an introduction to the subject
Member of
http://library.link/vocab/creatorName
Friedman, Robert
Dewey number
516.35
http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsaut
2F8BfX0jeEs
Image bit depth
0
Language note
English
LC call number
QA564-609
Literary form
non fiction
Nature of contents
dictionaries
Series statement
Universitext,
http://library.link/vocab/subjectName
  • Geometry, algebraic
  • Algebraic Geometry
Label
Algebraic Surfaces and Holomorphic Vector Bundles, by Robert Friedman, (electronic resource)
Instantiates
Publication
Note
Bibliographic Level Mode of Issuance: Monograph
Antecedent source
mixed
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Color
not applicable
Content category
text
Content type code
  • txt
Contents
1 Curves on a Surface -- Invariants of a surface -- Divisors on a surface -- Adjunction and arithmetic genus -- The Riemann-Roch formula -- Algebraic proof of the Hodge index theorem -- Ample and nef divisors -- Exercises -- 2 Coherent Sheaves -- What is a coherent sheaf? -- A rapid review of Chern classes for projective varieties -- Rank 2 bundles and sub-line bundles -- Elementary modifications -- Singularities of coherent sheaves -- Torsion free and reflexive sheaves -- Double covers -- Appendix: some commutative algebra -- Exercises -- 3 Birational Geometry -- Blowing up -- The Castelnuovo criterion and factorization of birational morphisms -- Minimal models -- More general contractions -- Exercises -- 4 Stability -- Definition of Mumford-Takemoto stability -- Examples for curves -- Some examples of stable bundles on ?2 -- Gieseker stability -- Unstable and semistable sheaves -- Change of polarization -- The differential geometry of stable vector bundles -- Exercises -- 5 Some Examples of Surfaces -- Rational ruled surfaces -- General ruled surfaces -- Linear systems of cubics -- An introduction toK3 surfaces -- Exercises -- 6 Vector Bundles over Ruled Surfaces -- Suitable ample divisors -- Ruled surfaces -- A brief introduction to local and global moduli -- A Zariski open subset of the moduli space -- Exercises -- 7 An Introduction to Elliptic Surfaces -- Singular fibers -- Singular fibers of elliptic fibrations -- Invariants and the canonical bundle formula -- Elliptic surfaces with a section and Weierstrass models -- More general elliptic surfaces -- The fundamental group -- Exercises -- 8 Vector Bundles over Elliptic Surfaces -- Stable bundles on singular curves -- Stable bundles of odd fiber degree over elliptic surfaces -- A Zariski open subset of the moduli space -- An overview of Donaldson invariants -- The 2-dimensional invariant -- Moduli spaces via extensions -- Vector bundles with trivial determinant -- Even fiber degree and multiple fibers -- Exercises -- 9 Bogomolov’s Inequality and Applications -- Statement of the theorem -- The theorems of Bombieri and Reider -- The proof of Bogomolov’s theorem -- Symmetric powers of vector bundles on curves -- Restriction theorems -- Appendix: Galois descent theory -- Exercises -- 10 Classification of Algebraic Surfaces and of Stable -- Bundles -- Outline of the classification of surfaces -- Proof of Castelnuovo’s theorem -- The Albanese map -- Proofs of the classification theorems for surfaces -- The Castelnuovo-deFranchis theorem -- Classification of threefolds -- Classification of vector bundles -- Exercises -- References
Dimensions
unknown
Edition
1st ed. 1998.
Extent
1 online resource (IX, 329 p.)
File format
multiple file formats
Form of item
online
Isbn
9781461216889
Level of compression
uncompressed
Media category
computer
Media type code
  • c
Other control number
10.1007/978-1-4612-1688-9
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000089656
  • (SSID)ssj0001295789
  • (PQKBManifestationID)11757436
  • (PQKBTitleCode)TC0001295789
  • (PQKBWorkID)11346606
  • (PQKB)10528956
  • (DE-He213)978-1-4612-1688-9
  • (MiAaPQ)EBC3077009
  • (EXLCZ)993400000000089656
Label
Algebraic Surfaces and Holomorphic Vector Bundles, by Robert Friedman, (electronic resource)
Publication
Note
Bibliographic Level Mode of Issuance: Monograph
Antecedent source
mixed
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Color
not applicable
Content category
text
Content type code
  • txt
Contents
1 Curves on a Surface -- Invariants of a surface -- Divisors on a surface -- Adjunction and arithmetic genus -- The Riemann-Roch formula -- Algebraic proof of the Hodge index theorem -- Ample and nef divisors -- Exercises -- 2 Coherent Sheaves -- What is a coherent sheaf? -- A rapid review of Chern classes for projective varieties -- Rank 2 bundles and sub-line bundles -- Elementary modifications -- Singularities of coherent sheaves -- Torsion free and reflexive sheaves -- Double covers -- Appendix: some commutative algebra -- Exercises -- 3 Birational Geometry -- Blowing up -- The Castelnuovo criterion and factorization of birational morphisms -- Minimal models -- More general contractions -- Exercises -- 4 Stability -- Definition of Mumford-Takemoto stability -- Examples for curves -- Some examples of stable bundles on ?2 -- Gieseker stability -- Unstable and semistable sheaves -- Change of polarization -- The differential geometry of stable vector bundles -- Exercises -- 5 Some Examples of Surfaces -- Rational ruled surfaces -- General ruled surfaces -- Linear systems of cubics -- An introduction toK3 surfaces -- Exercises -- 6 Vector Bundles over Ruled Surfaces -- Suitable ample divisors -- Ruled surfaces -- A brief introduction to local and global moduli -- A Zariski open subset of the moduli space -- Exercises -- 7 An Introduction to Elliptic Surfaces -- Singular fibers -- Singular fibers of elliptic fibrations -- Invariants and the canonical bundle formula -- Elliptic surfaces with a section and Weierstrass models -- More general elliptic surfaces -- The fundamental group -- Exercises -- 8 Vector Bundles over Elliptic Surfaces -- Stable bundles on singular curves -- Stable bundles of odd fiber degree over elliptic surfaces -- A Zariski open subset of the moduli space -- An overview of Donaldson invariants -- The 2-dimensional invariant -- Moduli spaces via extensions -- Vector bundles with trivial determinant -- Even fiber degree and multiple fibers -- Exercises -- 9 Bogomolov’s Inequality and Applications -- Statement of the theorem -- The theorems of Bombieri and Reider -- The proof of Bogomolov’s theorem -- Symmetric powers of vector bundles on curves -- Restriction theorems -- Appendix: Galois descent theory -- Exercises -- 10 Classification of Algebraic Surfaces and of Stable -- Bundles -- Outline of the classification of surfaces -- Proof of Castelnuovo’s theorem -- The Albanese map -- Proofs of the classification theorems for surfaces -- The Castelnuovo-deFranchis theorem -- Classification of threefolds -- Classification of vector bundles -- Exercises -- References
Dimensions
unknown
Edition
1st ed. 1998.
Extent
1 online resource (IX, 329 p.)
File format
multiple file formats
Form of item
online
Isbn
9781461216889
Level of compression
uncompressed
Media category
computer
Media type code
  • c
Other control number
10.1007/978-1-4612-1688-9
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000089656
  • (SSID)ssj0001295789
  • (PQKBManifestationID)11757436
  • (PQKBTitleCode)TC0001295789
  • (PQKBWorkID)11346606
  • (PQKB)10528956
  • (DE-He213)978-1-4612-1688-9
  • (MiAaPQ)EBC3077009
  • (EXLCZ)993400000000089656

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