Rigid Local Systems. (AM-139), Volume 139

Contents
Introduction
CHAPTER 1 First results on rigid local systems
	1.0 Generalities concerning rigid local systems over C
	1.1 The case of genus zero
	1.2 The case of higher genus
	1.3 The case of genus one
	1.4 The case of genus one: detailed analysis
CHAPTER 2 The theory of middle convolution
	2.0 Transition from irreducible local systems on open sets of P1 to irreducible middle extension sheaves on A1
	2.1 Transition from irreducible middle extension sheaves on A1 to irreducible perverse sheaves on A1
	2.2 Review of Dbc(X, Ql)
	2.3 Review of perverse sheaves
	2.4 Review of Fourier Transform
	2.5 Review of convolution
	2.6 Convolution operators on the category of perverse sheaves: middle convolution
	2.7 Interlude: middle direct images (relative dimension one)
	2.8 Middle additive convolution via middle direct image
	2.9 Middle additive convolution with Kummer sheaves
	2.10 Interpretation of middle additive convolution via Fourier Transform
	2.11 Invertible objects on A1 in characteristic zero
	2.12 Musings on *mid -invertible objects in P in the Gm case
	2.13 Interlude: surprising relations between *mid on A^1 and on Gm
	2.14 Interpretive remark: Fourier-Bessel Transform
	2.15 Questions about the situation in several variables
	2.16 Questions about the situation on elliptic curves
	2.17 Appendix 1: the basic lemma on end-exact functors
	2.18 Appendix 2: twisting representations by characters
CHAPTER 3 Fourier Transform and rigidity
	3.0 Fourier Transform and index of rigidity
	3.1 Lemmas on representations of inertia groups
	3.2 Interlude: the operation ⊗mid
	3.3 Applications to middle additive convolution
	3.4 Some open questions about local Fourier Transform
CHAPTER 4 Middle convolution: dependence on parameters
	4.0 Good schemes
	4.1 The basic setting
	4.2 Basic results in the basic setting
	4.3 Middle convolution in the basic setting
CHAPTER 5 Structure of rigid local systems
	5.0 Cohomological rigidity
	5.1 The category Tl, and the functors MCχ and MTℒ
	5.2 The main theorem on the structure of rigid local systems
	5.3 Applications and Interpretations of the main theorem
	5.4 Some open questions
	5.5 Existence of universal families of rigids with given local monodromy
	5.6 Remark on braid groups
	5.7 Universal families without quasiunipotence
	5.8 The complex analytic situation
	5.9 Return to the original question
CHAPTER 6 Existence algorithms for rigids
	6.0 Numerical invariants
	6.1 Numerical incarnation: the group NumData
	6.2 A compatibility theorem
	6.3 Realizable and plausible elements
	6.4 Existence algorithm for
rigids
	6.5 An example
	6.6 Open questions
	6.7 Action of automorphisms
	6.8 A remark and a question
CHAPTER 7 Diophantine aspects of
rigidity
	7.0 Diophantine criterion for irreducibility
	7.1 Diophantine criterion for rigidity
	7.2 Appendix: a counterexample
CHAPTER 8 Motivic description of
rigids
	8.0 The basic setting
	8.1 Interlude: Kummer sheaves
	8.2 Naive convolution on (A1- {T1, ... , Tn})SN,n,l'
	8.3 Middle convolution on (A1- {T1, ... , Tn})SN,n,l'
	8.4 "Geometric" description of all tame rigids with quasi-unipotent local monodromy
	8.5  A remark and a question
CHAPTER 9 Grothendieck’s p-curvature conjecture for
rigids
	9.0 Introduction
	9.1 Review of Grothendieck’s p-curvature
conjecture
	9.2 Interlude: Picard-Fuchs equations and some variants
	9.3 The main result of [Ka-ASDE] and a generalization
	9.4 Application to rigid local systems
	9.5 Comments and questions
References