Weil Conjectures, Perverse Sheaves and I'Adic Fourier Transform

by Kiehl, Reinhardt; Weissauer, R.; Feferman, S.; Gromov, M.; Jost, J.; Kollar, J.; Lenstra, H. W., Jr.

• ISBN13: 9783540414575
• ISBN10: 3540414576
• Format: Hardcover
• Copyright: 2001-11-01
• Publisher: Springer Verlag

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