代數曲面和全純向量叢（簡體書）

《代數曲面和全純向量叢(英文版)》主要內容包括：An Introduction to Elliptic Surfaces、Singular fibers、Singulex fibers of elliptic fibrations、lnvariants and the canonical bundle formula、Elliptic surfaces with a section and Weierstrass models、More general elliptic surfaces、The fundamental group等。本書是一本經典的的大學生數學教程，綜合講述了代數面理論和正則向量叢。定位於對代數幾何有初步瞭解的研究生以及代數幾何、規範理論和4-流形拓撲領域的研究生和研究人員。許多弦理論方面的結果是物理學家是非常感興趣的。本書綜合講述代數面理論，對線和麵上的向量叢理論的研究。在書的前幾章面理論和向量叢的講述是單獨進行的，不同的章節進行研究。隨著書中內容的深入，它們的相互滲透作用顯得越來越明顯。詳細討論了直紋面和橢圓面，進而分析了面上的向量叢幾何。向量叢的許多結果第一次以書的形式出現，很適合研究生學習。本書注重實際解決問題的能力，100多道練習題是不可缺少的組成部分。目次：面上的曲線；凝聚層；雙有理幾何；穩定性；有關面的例子；直紋面上的向量叢；橢圓面導論；橢圓面上的向量叢；Bogomolov不等式和應用；代數面的分類和穩定叢。
讀者對象：適用于數學專業本科生、研究生以及自學的相關人員。

Preface
Introduction
1 Curves on a Surface
Introduction
Invariants of a surface
Divisors on a surface
Adjunetion 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 B|ratlonal Geometry
Blowing up
The Castelnuovo criterion and factorization of birationa] morphisms
Minimal models
More general contractions
Exercises

4 Stability
Definition of Mumford-Takemoto stability
Examples for curves
Some examples of stable bundles on p2
Gieseker stability
Unstable and semlstable 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 to K3 surfaces
Exercises

6 Vector Bundles over Ruled Surfaces
Suitable ample divisors
Ruled sur faces
A brief introduction to local and global moduli
A Zariski open subeet of the moduli space
Exercises

7 An Introduction to Elliptic Surfaces
Singular fibers
Singulex fibers of elliptic fibrations
lnvariants 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 elliptie surface*
A Zariski open subset of the modnii space
An overview of Donaldson invariants
The 2-dimensional invariant
Moduli spaces via extensions
Vector bundleswi thtrivial determinant
Even fiber degree and multiple fibers
Exercises

9 Bogomolovs Inequality and Applications
Statementofththeorem
The theorems of Bombieri and Reider
Theproof of Bogomolovs theorem
Symmetric powers of vector bundles on curves
Restriction the orems
Appendix:Galoisdes centtheory
Exercises

10 Classification of Algebraic Surfacesand of Stable Bundles
Outline of the classificationofs urfaces
Proof of Castelnuovos theorem
The Albanese map
Proofs of the classification the oremsf or surfaces
The Castelnuovo-deFranchis theorem
Classification of threefolds
Classification of vector bundles
Exercises
References
Index