Complex Analytic Geometry: From the Localization Viewpoint

Contents
Preface
About the Author
1. Analytic Functions of Several Complex Variables
1.1 Multiseries
1.2 Analytic functions of one complex variable
1.3 Analytic functions of several complex variables
1.4 Germs of holomorphic functions
2. Complex Manifolds and Analytic Varieties
2.1 Complex manifolds
2.2 Analytic varieties
2.3 Germs of varieties
2.4 Nullstellensatz and dimension
2.5 Underlying real structures
3. Vector Bundles
3.1 Group actions
3.2 Fiber bundles
3.3 Vector bundles
3.4 Tangent bundle and vector fields
3.5 Stiefel manifold
3.6 Grassmann manifold
3.7 Some topics on differentiable manifolds
4. Dualities and Thom Class
4.1 Algebraic topology on manifolds
4.2 Poincaré, Alexander and Lefschetz dualities
4.3 Thom isomorphism and Thom class
4.4 Intersection product
5. Chern Classes and Localization via Obstruction Theory
5.1 Index of a family of sections
5.2 Chern classes of a complex vector bundle
5.3 Euler class of an oriented real vector bundle
5.4 Relative classes
5.5 Piecewise-linear manifolds and pseudo-manifolds
5.6 Localization and topological residues
6. Differential Forms
6.1 Vector fields and differential forms
6.2 Integration of differential forms
6.3 Integration along fibers
6.4 Frobenius theorem and non-singular foliations
7. Čech-de Rham Cohomology
7.1 de Rham cohomology
7.2 Čech cohomology
7.3 Čech-de Rham cohomology
7.4 Integration of Čech-de Rham cochains
7.5 Combination with combinatorial topology
7.6 de Rham theorem and related topics
7.7 Relative Čech-de Rham cohomology
7.8 Description of dualities in differential forms
7.9 Thom class in Čech-de Rham cohomology
7.10 Angular form and Bochner-Martinelli form
7.11 Lefschetz fixed point formula
8. Chern-Weil Theory Adapted to Cech-de Rham Cohomology
8.1 Connections
8.2 Characteristic classes of complex vector bundles
8.3 Further topics on connections
8.4 Characteristic classes in Čech-de Rham cohomology
9. Vector Bundles with Metrics and Related Topics
9.1 Metrics on vector bundles and harmonic forms
9.2 Hodge structures
9.3 Vector bundles on projective space
9.4 Kähler manifolds
9.5 Atiyah classes
9.6 Hermitian connections and Bott-Chern classes
10. Localization
10.1 Localization and associated residues
10.2 Vanishing theorems
10.3 Differential geometric localization by frames
10.4 Thom class of a complex vector bundle
10.5 Coincidence of topological and differential geometric localizations
11. Further Topics
11.1 Sheaves
11.2 Sheaf cohomology
11.3 Coherent sheaves
11.4 Dolbeault theorem
11.5 Complex analytic spaces
11.6 Divisors
11.7 Local complete intersections
12. Residues of Chern Classes on Manifolds
12.1 Triangulation of subanalytic sets
12.2 Residues of Chern classes on manifolds
12.3 Grothendieck residues
12.4 Residues at an isolated singularity
12.5 Examples
12.6 Dual class of a complex subspace
13. Residues of Chern Classes on Singular Varieties
13.1 Controlled tube systems for Whitney stratifications
13.2 Poincaré, Alexander, Lefschetz and Thom morphisms
13.3 de Rham and Čech-de Rham theories for singular varieties
13.4 Chern classes and localization
13.5 Residues of Chern classes on singular varieties
13.6 Residues at an isolated singularity
13.7 Examples and related topics
14. Intersection Product of Complex Subspaces
14.1 Refined Whitney sum formula
14.2 Intersection product in complex manifolds
14.3 Intersection product in singular varieties
14.4 Intersection product in singular surfaces
14.5 Excess intersections
15. Riemann-Roch Theorem
15.1 Riemann-Roch problem for curves
15.2 Characteristic classes of virtual bundles
15.3 Chern-Weil theory for virtual bundles
15.4 Local Chern classes and characters
15.5 Universal localized Riemann-Roch theorem
15.6 Riemann-Roch theorem for embeddings
15.7 Grothendieck-Riemann-Roch Theorem
Appendix A Commutative Algebra
A.1 Homological algebra
A.2 Commutative rings
Notes
Appendix B Algebraic Topology
B.1 Singular homology
B.2 Cell complexes
B.3 Homology of locally finite chains
Bibliography
Index