Introduction

A Short History：Les débuts de la théorie des faisceaux by Christian Houzel

Ⅰ．Homological algebra

Summary

1.1．Categories and functors

1.2．Abelian categories

1.3．Categories of complexes

1.4．Mapping cones

1.5．Triangulated categories

1.6．Localization of categories

1.7．Derived categories

1.8．Derived functors

1.9．Double complexes

1.10．Bifunctors

1.11．Ind-objects and pro-objects

1.12．The Mittag-Leffler condition

Exercises to Chapter Ⅰ

Notes

Ⅱ．Sheaves

Summary

2.1．Presheaves

2.2．Sheaves

2.3．Operations on sheaves

2.4．Injective,flabby and flat sheaves

2.5．Sheaves on locally compact spaces

2.6．Cohomology of sheaves

2.7．Some vanishing theorems

2.8．Cohomology of coverings

2.9．Examples of sheaves on real and complex manifolds

Exercises to Chapter Ⅱ

Notes

Ⅲ．Poincaré-Verdier duality and Fourier-Sato transformation

Summary

3.1．Poincaré-Verdier duality

3.2．Vanishing theorems on manifolds

3.3．Orientation and duality

3.4．Cohomologically constructible sheaves

3.5．γ-topology

3.6．Kernels

3.7．Fourier-Sato transformation

Exercises to Chapter Ⅲ

Notes

Ⅳ．Specialization and microlocalization

Summary

4.1．Normal deformation and normal cones

4.2．Specialization

4.3．Microlocalization

4.4．The functor μhom

Exercises to Chapter Ⅳ

Notes

Ⅴ．Micro-support of sheaves

Summary

5.1．Equivalent definitions of the micro-support

5.2．Propagation

5.3．Examples：micro-supports associated with locally closed subscts

5.4．Functorial properties of the micro-support

5.5．Micro-support of conic sheaves

Exereises to Chapter Ⅴ

Notes

Ⅵ．Micro-support and microlocalization

Summary

6.1．The category Db（X；Ω）

6.2．Normal cones in cotangent bundles

6.3．Direct images

6.4．Microlocalization

6.5．Involutivity and propagation

6.6．Sheaves in a neighborhood of an involutive manifold

6.7．Microlocalization and inverse images

Exercises to Chapter Ⅵ

Notes

Ⅶ．Contact transformations and pure sheaves

Summary

7.1．Microlocal kernels

7.2．Contact transfornations for sheaves

7.3．Microlocal composition of kernels

7.4．Integral transformations for sheaves associated with submanifolds

7.5．Pure sheaves

Exercises to Chapter Ⅶ

Notes

Ⅷ．Constructible sheaves

Summary

8.1．Constructible sheaves on a simplicial complex

8.2．Subanalytic sets

8.3．Subanalytic isotropic sets and μ-stratifications

8.4．R-constructible sheaves

8.5．C-constructible sheaves

8.6．Nearby-cycle functor and vanishing-cycle functor

Exercises to Chapter Ⅷ

Notes

Ⅸ．Characteristic cycles

Summary

9.1．Index formula

9.2．Subanalytic chains and subanalytic cycles

9.3．Lagrangian cycles

9.4．Characteristic cycles

9.5．Microlocal index formulas

9.6．Lefschetz fixed point formula

9.7．Constructible functions and Lagrangian cycles

Exercises to Chapter Ⅸ

Notes

Ⅹ．Perverse sheaves

Summary

10.1．t-structures

10.2．Perverse sheaves on real manifolds

10.3．Perverse sheaves on complex manifolds

Exercises to Chapter Ⅹ

Notes

Ⅺ．Applications to O-modules and D-modules

Summary

11.1．The sheaf Ox

11.2．Dx-modules

11.3．Holomorphic solutions of Dx-modules

11.4．Microlocal study of Ox

11.5．Microfunctions

Exercises to Chapter Ⅺ

Notes

Appendix：Symplectic geometry

Summary

A.1．Symplectic vector spaces

A.2．Homogeneous symplectic manifolds

A.3．Inertia index

Exercises to the Appendix

Notes

Bibliography

List of notations and conventions

Index