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＜電子ブック＞
Algebraic Surfaces and Holomorphic Vector Bundles / by Robert Friedman
(Universitext. ISSN:21916675)

版	1st ed. 1998.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	1998
本文言語	英語
大きさ	IX, 329 p : online resource
著者標目	*Friedman, Robert author SpringerLink (Online service)
件　名	LCSH:Algebraic geometry FREE:Algebraic Geometry
一般注記	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 ruledsurfaces -- 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 ofthe 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 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 ofbundles 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 HTTP:URL=https://doi.org/10.1007/978-1-4612-1688-9

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