Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman (English) P

Description: Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman 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. FORMAT Paperback LANGUAGE English CONDITION Brand New Publisher Description 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time, the focus of 4-manifold topology has shifted dramatically, at first be cause topological methods have largely superseded algebro-geometric meth ods in computing Donaldson invariants, and more importantly because of and Witten, which have greatly sim the new invariants defined by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. Table of 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 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 Bogomolovs Inequality and Applications.- Statement ofthe theorem.- The theorems of Bombieri and Reider.- The proof of Bogomolovs 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 Castelnuovos theorem.- The Albanese map.- Proofs of the classification theorems for surfaces.- The Castelnuovo-deFranchis theorem.- Classification of threefolds.- Classification of vector bundles.- Exercises.- References. Promotional Springer Book Archives Long Description This book is based on courses given at Columbia University on vector bun Feature One of the books primary assets is its method of presentation which makes the subject rather accessible The only prerequisite is a good working knowledge of elementary algebraic geometry Unified introduction to the study of algebraic surfaces and vector bundles Algebraic geometry is an active area of current research Description for Sales People The goal of this book is to provide a unified introduction to the study of algebraic surfaces and of holomorphic vector bundles on them. The author keeps the prerequisites to a good working knowledge of Hartshornes book on algebraic geometry. The book will appeal to graduate students and researchers in the areas of algebraic geometry, gauge thoery, or 4-manifold topolgogy, as well as to physicists studying string theory. Details ISBN1461272467 Author Robert Friedman Language English Edition 98199th ISBN-10 0387983619 Short Title ALGEBRAIC SURFACES & HOLOMORPH Series Universitext Media Book Affiliation Clinical Assistant Professor Department of Dermatology New York Univer DEWEY 516.35 Year 2012 Publication Date 2012-10-08 Imprint Springer-Verlag New York Inc. Place of Publication New York, NY Country of Publication United States Illustrations IX, 329 p. AU Release Date 2012-10-08 NZ Release Date 2012-10-08 US Release Date 2012-10-08 UK Release Date 2012-10-08 Pages 329 Publisher Springer-Verlag New York Inc. Edition Description Softcover reprint of the original 1st ed. 1998 ISBN-13 9781461272465 Format Paperback Alternative 9780387983615 Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. With fast shipping, low prices, friendly service and well over a million items - you're bound to find what you want, at a price you'll love! 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