基本説明
Contents: Curves on a Surface; Coherent Sheaves; Birational Geometry; Stability; Some examples of surfaces; Vector Bundles over Ruled Surfaces; An Introduction to Elliptic Surfaces; Vector Bundles over Elliptic Surfaces, and more.
Full 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time, the focus of 4-manifold topology has shifted dramatically, at first be cause topological methods have largely superseded algebro-geometric meth ods in computing Donaldson invariants, and more importantly because of and Witten, which have greatly sim the new invariants defined by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.
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 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.
