Author | Friedman, Robert. author |
---|---|
Title | Algebraic Surfaces and Holomorphic Vector Bundles [electronic resource] / by Robert Friedman |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1998 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1688-9 |
Descript | IX, 329 p. online resource |
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 Bogomolovโs Inequality and Applications -- Statement of the theorem -- The theorems of Bombieri and Reider -- The proof of Bogomolovโs 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 Castelnuovoโs theorem -- The Albanese map -- Proofs of the classification theorems for surfaces -- The Castelnuovo-deFranchis theorem -- Classification of threefolds -- Classification of vector bundles -- Exercises -- References