Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform

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