文档介绍：Algebra and Geometry in several
Complex variables
Prof.
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Contents
Chapter 1. Holomorphic functions of several variables
Analysis plex plane
Holomorphic functions of several variables
Hartogs’ phenomenon
Differential forms
Bochner-Martinelli formula
Analytic continuation from boundary
Problems
Leray’ formula
Chapter 2. Analytic manifolds
Submanifolds
Simplectic structure
Analytic manifolds
Chapter 3. Division of holomorphic functions
Weierstrass theorem
Stabilization and finiteness properties
Chapter 4. Analytic sets
Analytic sets and germs
Resultant
Discriminant
Hypersurfaces
Geometry of irreducible germs
Continuity of roots of a polynomial
Chapter 5. Analytic and polynomial algebras
Local and global algebras
Primary position
intersection ideals
Zero-dimensional ideals
Classical theory of polynomial ideals
2
Chapter 6. N¨other operators, residue and bases
Differential operators in modules
Polynomial ideals revisited
Residue
Linear bases in ideals
Bases and division in a module
Chapter 7. Sheaves
Categories and functors
Abelian categories
Sheaves
Chapter 8. Coherent sheaves and analytic spaces
Analytic sheaves
Coherent sheaves of ideals
Category of analytic algebras
analytic spaces
Fibre products
Chapter 9. Elements of homological algebra
and homology
Exact functors
Tensor products
Projective resolvents
Chapter 10. Derived functors, Ext and Tor
Derived functors
Examples
Exact sequence
Properties of Tor
Flat modules
Syzygy theorem
Chapter 11. Deformation of analytic spaces
Criterium of flatness
Examples
Properties
Deformation of analytic spaces
3
Chapter 12. Finite morphisms
Direct image
Multiplicity of flat morphisms
Example