Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform

✍ Scribed by Reinhardt Kiehl

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 382
• Series: Ergebnisse der Mathematik und ihrer Grenzgebiete 42
• Category: Library

✦ Table of Contents

Preface
Table of Contents
Frequently Used Notation
Introduction
I. The General Weil Conjectures (Deligne's Theory of Weights)
I.1 Weil Sheaves
I.2 Weights
I.3 The Zariski Closure of Monodromy
I.4 Real Sheaves
I.5 Fourier Transform
I.6 Weil Conjectures (Curve Case)
I.7 The Weil Conjectures for a Morphism (General Case)
I.8 Some Linear Algebra
I.9 Refinements (Local Monodromy)
II. The Formalism of Derived Categories
II.1 Triangulated Categories
II.2 Abstract Truncations
II.3 The Core of a t-Structure
II.4 The Cohomology Functors
II.5 The Triangulated Category D^b_c (X, \bar{Q}_l )
5.1 Technical Remarks
II.6 The Standard t-Structure on D^b_c (X, o)
Deligne's Truncation Operation
The Deligne Operator
II.7 Relative Duality for Singular Morphisms
II.8 Duality for Smooth Morphisms
II.9 Relative Duality for Closed Embeddings
II.10 Proof of the Biduality Theorem
II.11 Cycle Classes
The Chern Class
An Alternative Definition
II.12 Mixed Complexes
III. Perverse Sheaves
III.1 Perverse Sheaves
III.2 The Smooth Case
III.3 Glueing
Truncation of Mixed Perverse Sheaves
III.4 Open Embeddings
III.5 Intermediate Extensions
III.6 Affine Maps
III.7 Equidimensional Maps
III.8 Fourier Transform Revisited
III.9 Key Lemmas on Weights
III.10 Gabber's Theorem
III.11 Adjunction Properties
III.12 The Dictionary
III.13 Complements on Fourier Transform
III.14 Sections
III.15 Equivariant Perverse Sheaves
III.16 Kazhdan-Lusztig Polynomials
Correspondences on B
Another Description
The Hecke Ring H
Verdier Duality
Purity Properties
IV. Lefschetz Theory and the Brylinski-Radon Transform
IV.1 The Radon Transform
Constant Sheaves
Radon Inversion
IV.2 Modified Radon Transforms
Quotient Categories
Extensions
The Proofs
IV.3 The Universal Chern Class
3.1 Cup Product
3.2 A Factorization
3.3 Fact
IV.4 Hard Lefschetz Theorem
IV.5 Supplement: A Spectral Sequence
V. Trigonometric Sums
V.1 Introduction
V.2 Generalized Kloosterman Sums
V.3 Links with l-adic Cohomology
V.4 Deligne's Estimate
V.5 The Swan Conductor
V.6 The Ogg-Shafarevich-Grothendieck Theorem
V.7 The Main Lemma
V.8 The Relative Abhyankar Lemma
V.9 Proof of the Theorem of Katz
V.10 Uniform Estimates
V.11 An Application
Bibliography for Chapter V
VI. The Springer Representations
VI.1 Springer Representations of Weyl Groups of SemisimpleAlgebraic Groups
VI.2 The Flag Variety B
VI.3 The Nilpotent Variety N
VI.4 The Lie Algebra in Positive Characteristic
VI.5 Invariant Bilinear Forms on g
VI.6 The Normalizer of Lie(B)
VI.7 Regular Elements of the Lie Algebra g
VI.8 Grothendieck's Simultaneous Resolution of Singularities
The Adjoint Representation
The Weyl Group
Adjoint Quotients
VI.9 The Galois Group W
VI.10 The Monodromy Complexes Φ and Φ'
VI.11 The Perverse Sheaf Ψ
VI.12 The Orbit Decomposition of Ψ
VI.13 Proof of Springer's Theorem
VI.14 A Second Approach
VI.15 The Comparison Theorem
VI.16 Regular Orbits
VI.17 W-actions on the Universal Springer Sheaf
VI.18 Finite Fields
Concluding Remarks
VI.19 Determination of ε_T
Bibliography for Chapter VI
Appendix
A. \hat{Q}_l-Sheaves
B. Bertini Theorem for Etale Sheaves
C. Kummer Extensions
D. Finiteness Theorems
Vanishing Cycles
Construction of the Variation map
Constructibility of RΨ(K^●) and RΦ(K^●)
Passage to l-adic Cohomology
Bibliography
1-23
24-48
49-73
74-93
94-119
120-139
140-161
162-186
187-212
213-238
239-265
266-291
292-316
317-324
Glossary
Index