De Rham Cohomology of Differential Modules on Algebraic Varieties (Progress in Mathematics (189), Band 189)

Contents
Introduction
Chapter I Differential algebra
	Introduction
	1 Hypergeometric origins
		1.1 Gauss hypergeometric differential equation
		1.2 Kummer confluent hypergeometric differential equation
	2 From differential equations to differential modules
		2.1 Derivations and differentials
		2.2 Differential rings
		2.3 Equivalence of differential systems
		2.4 Differential modules
		2.5 Solutions in a differential extension. Duality
		2.6 Relation between differential modules and differential systems
		2.7 Tensor product and related operations
		2.8 Trace morphism
	3 Back to differential equations: cyclic vectors
		3.1 Differential operators
		3.2 Cyclic vectors
		3.3 Construction of cyclic vectors
Chapter II Connections on algebraic varieties
	Introduction
	4 Connections
		4.1 Differential forms and jets
		4.2 Connections
		4.3 Integrable connections and de Rham complexes
		4.4 Relation to differential modules and differential systems
		4.5 Connections on vector bundles
		4.6 Cyclic vectors
	5 Inverse and direct images
		5.1 Inverse image
		5.2 Direct image by an étale morphism
Chapter III Regularity: formal theory
	Introduction
	6 Hypergeometric equations
		6.1 Singular points of hypergeometric equations
		6.2 Local monodromy
		6.3 Fuchs-Frobenius theory
	7 The classical formal theory of regular singular points
		7.1 The exponential formalism xA
		7.2 Non-resonance
		7.3 Indicial polynomials
		7.4 Regularity of differential systems
		7.5 Regularity criterion for differential equations
		7.6 Exponents
	8 Jordan decomposition of differential modules
		8.1 Jordan theory for differential modules
		8.2 Action of commuting derivations
		8.3 The regular case
		8.4 Variant with parameters
	9 Formal integrable connections (several variables)
		9.1 Outline of Gérard-Levelt theory
		9.2 Regularity and logarithmic extensions
Chapter IV Regularity: geometric theory
	Introduction
	10 Regularity and exponents along prime divisors
		10.1 Transversal derivations and integral curves
		10.2 Regular connections along prime divisors
		10.3 Exponents along prime divisors
	11 Regularity and exponents along a normal crossing divisor
		11.1 Connections with logarithmic poles, and residues
		11.2 Extensions with logarithmic poles
		11.3 On reflexivity
		11.4 Construction (and uniqueness) of
		11.5 Local freeness of M
	12 Base change
		12.1 Restriction to curves I. The case when C meets D transversally at a smooth point
		12.2 Restriction to curves II. The case when D is a strict normal crossing divisor
		12.3 Restriction to curves III. The general case
		12.4 Pull-back of a regular connection along D
	13 Global regularity and exponents
		13.1 Global regularity
		13.2 Global exponents
Chapter V Irregularity: formal theory
	Introduction
	14 Confluent hypergeometric equations and phenomena related to irregularity
		14.1 Solutions of the confluent hypergeometric equation
		14.2 Meromorphic coefficients and Stokes multipliers
	15 Poincaré rank
		15.1 Spectral norms
		15.2 Christol-Dwork-Katz theorem
		15.3 Poincaré rank
	16 Turrittin-Levelt decomposition and variants
		16.1 The Turrittin-Levelt decomposition
		16.2 Proof of the decomposition
	17 Slopes and Newton polygons
		17.1 Slope decomposition
		17.2 Newton polygons
		17.3 Newton polygons of cyclic modules
		17.4 Index of operators and Malgrange's definition of irregularity
		17.5 Variant with parameters. Turning points
		17.6 Variation of the Newton polygon
	18 Varia
		18.1 Cyclic vectors in the neighborhood of a non-turning singular point
		18.2 Turrittin decomposition around crossing points of the polar divisor
Chapter VI Irregularity: geometric theory
	Introduction
	19. Poincaré rank and Newton polygon (prime divisor).
		19.1 Poincaré rank along a prime divisor
		19.2 Newton polygon along a prime divisor
		Stratificat19.3 ion of the polar divisor by Newton polygons
	20 Turrittin-Levelt decomposition and -extensions
		20.1 Formal Turrittin decomposition along a divisor
		20.2 -extensions of irregular connections
	21 Main theorem on the Poincaré rank
		21.1 Statement of the main theorem
		21.2 Proof of the main theorem
Chapter VII de Rham cohomology and Gauss-Manin connection
	Introduction
	22 Hypergeometric equation and Euler representation
	23 de Rham cohomology and the Gauss-Manin connection
		23.1 Direct image and higher direct images
		23.2 de Rham and Spencer complexes
		23.3 Some spectral sequences
		23.4 Local construction of the Gauss-Manin connection
		23.4 Flat base change
		23.6 Vanishing and computation
	24 Index formula
		24.1 Deligne's global index formula on algebraic curves
		24.2 Proof of the global index formula
Chapter VIII Elementary fibrations and applications
	Introduction
	25 Elementary fibrations and dévissage
		25.1 Elementary fibrations
		25.2 Artin sets
		25.3 Dévissage
	26 Main theorems on the Gauss-Manin connection
		26.1 Generic finiteness of direct images
		26.2 Generic base change for direct images
	27 Gauss-Manin connection in the regular case
		27.1 Main theorems (in the regular case)
		27.2 Coherence of the cokernel of a regular connection
		27.3 Regularity and exponents of the cokernel of a regular connection
Chapter IX Complex and p-adic comparison theorems
	Introduction
	28 The hypergeometric situation
	29 Analytic contexts
		29.1 Complex-analytic connections
		29.2 Rigid analytic connections
	30 Abstract comparison criteria
		30.1 First criterion
		30.2 Second criterion
	31 Comparison theorem for algebraic vs. complex-analytic cohomology
		31.1 Statement of the comparison
		31.2 Reduction to the case of a rational elementary fibration
		31.3 First way: reduction to an ordinary linear differential system
		31.4 Second way: dealing with the relative situation
		31.5 Deligne's GAGA version of the index formula
	32 Comparison theorem for algebraic vs. rigid-analytic (regular coefficients)
		32.1 Liouville numbers
		32.2 Comparison
	33 Rigid-analytic comparison theorem in relative dimension one
		33.1 On the coherence of the cokernel of a connection in the rigid analytic situation
		33.2 Rigid analytic comparison theorem in relative dimension one
	34 Comparison theorem for algebraic vs. rigid-analytic (irregular coefficients)
		34.1 Statement
		34.2 Key propositions
		34.3 Proof
		34.4 Proof of 34.2.1
		34.5 Proof of 34.2.2
		34.6 Properties of the GAGA functor
Appendix A Riemann's existence theorem" in higher dimension, an elementary approach
Bibliography
Index