Weil Conjectures, Perverse Sheaves and I'Adic Fourier Transform,9783540414575

Weil Conjectures, Perverse Sheaves and I'Adic Fourier Transform

by ; ; ; ; ; ;
ISBN13: 9783540414575
ISBN10: 3540414576
Format: Hardcover
Pub. Date: 2001-11-01
Publisher(s): Springer Verlag
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Summary

In this book the authors describe the important generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La conjecture de Weil II". The authors follow the important and beautiful methods of Laumon and Brylinski which lead to a simplification of Deligne's theory. Deligne's work is closely related to the sheaf theoretic theory of perverse sheaves. In this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore the authors include the complete theory of middle perverse sheaves. In this part, the l-adic Fourier transform is introduced as a technique providing natural and simple proofs. To round things off, there are three chapters with significant applications of these theories.

Table of Contents

Introduction 1(4)
The General Weil Conjectures (Deligne's Theory of Weights)
5(62)
Weil Sheaves
5(8)
Weights
13(12)
The Zariski Closure of Monodromy
25(8)
Real Sheaves
33(5)
Fourier Transform
38(7)
Weil Conjectures (Curve Case)
45(7)
The Weil Conjectures for a Morphism (General Case)
52(2)
Some Linear Algebra
54(4)
Refinements (Local Monodromy)
58(9)
The Formalism of Derived Categories
67(68)
Triangulated Categories
67(7)
Abstract Truncations
74(3)
The Core of a t-Structure
77(4)
The Cohomology Functors
81(5)
The Triangulated Category Dbc(X, Ql)
86(12)
The Standard t-Structure on Dbc(X, o)
98(8)
Relative Duality for Singular Morphisms
106(6)
Duality for Smooth Morphisms
112(4)
Relative Duality for Closed Embeddings
116(3)
Proof of the Biduality Theorem
119(4)
Cycle Classes
123(6)
Mixed Complexes
129(6)
Perverse Sheaves
135(68)
Perverse Sheaves
135(2)
The Smooth Case
137(2)
Glueing
139(5)
Open Embeddings
144(3)
Intermediate Extensions
147(6)
Affine Maps
153(3)
Equidimensional Maps
156(3)
Fourier Transform Revisited
159(2)
Key Lemmas on Weights
161(6)
Gabber's Theorem
167(2)
Adjunction Properties
169(4)
The Dictionary
173(4)
Complements on Fourier Transform
177(4)
Sections
181(2)
Equivariant Perverse Sheaves
183(6)
Kazhdan-Lusztig Polynomials
189(14)
Lefschetz Theory and the Brylinski-Radon Transform
203(22)
The Radon Transform
203(4)
Modified Radon Transforms
207(8)
The Universal Chern Class
215(2)
Hard Lefschetz Theorem
217(4)
Supplement: A Spectral Sequence
221(4)
Trigonometric Sums
225(24)
Introduction
225(1)
Generalized Kloosterman Sums
226(3)
Links with l-adic Cohomology
229(1)
Deligne's Estimate
230(1)
The Swan Conductor
231(5)
The Ogg-Shafarevich-Grothendieck Theorem
236(1)
The Main Lemma
237(3)
The Relative Abhyankar Lemma
240(1)
Proof of the Theorem of Katz
241(3)
Uniform Estimates
244(2)
An Application
246(3)
Bibliography for Chapter V
248(1)
The Springer Representations
249(74)
Springer Representations of Weyl Groups of Semisimple Algebraic Groups
249(4)
The Flag Variety B
253(3)
The Nilpotent Variety N
256(5)
The Lie Algebra in Positive Characteristic
261(2)
Invariant Bilinear Forms on g
263(1)
The Normalizer of Lie(B)
264(1)
Regular Elements of the Lie Algebra g
264(2)
Grothendieck's Simultaneous Resolution of Singularities
266(3)
The Galois Group W
269(3)
The Monodromy Complexes Φ and Φ
272(4)
The Perverse Sheaf ψ
276(2)
The Orbit Decomposition of ψ
278(3)
Proof of Springer's Theorem
281(5)
A Second Approach
286(4)
The Comparison Theorem
290(5)
Regular Orbits
295(6)
W-actions on the Universal Springer Sheaf
301(9)
Finite Fields
310(7)
Determination of εT
317(6)
Bibliography for Chapter VI
319(4)
Appendix 323(32)
A. Ql-Sheaves
323(10)
B. Bertini Theorem for Etale Sheaves
333(3)
C. Kummer Extensions
336(2)
D. Finiteness Theorems
338(17)
Bibliography 355(16)
Glossary 371(2)
Index 373

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