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

"...A nice feature of the book [is] that at various points the authors provide examples, or rather counterexamples, that clearly show what can go wrong...This is a nicely-written book [that] studies algebraic differential modules in several variables." --Mathematical Reviews Contents 6 Introduction 11 Chapter I Differential algebra 15 Introduction 15 1 Hypergeometric origins 15 1.1 Gauss hypergeometric differential equation 15 1.2 Kummer confluent hypergeometric differential equation 16 2 From differential equations to differential modules 17 2.1 Derivations and differentials 17 2.2 Differential rings 18 2.3 Equivalence of differential systems 19 2.4 Differential modules 20 2.5 Solutions in a differential extension. Duality 21 2.6 Relation between differential modules and differential systems 23 2.7 Tensor product and related operations 24 2.8 Trace morphism 25 3 Back to differential equations: cyclic vectors 27 3.1 Differential operators 27 3.2 Cyclic vectors 29 3.3 Construction of cyclic vectors 33 Chapter II Connections on algebraic varieties 37 Introduction 37 4 Connections 37 4.1 Differential forms and jets 37 4.2 Connections 39 4.3 Integrable connections and de Rham complexes 41 4.4 Relation to differential modules and differential systems 42 4.5 Connections on vector bundles 43 4.6 Cyclic vectors 43 5 Inverse and direct images 44 5.1 Inverse image 45 5.2 Direct image by an étale morphism 46 Chapter III Regularity: formal theory 48 Introduction 48 6 Hypergeometric equations 48 6.1 Singular points of hypergeometric equations 48 6.2 Local monodromy 50 6.3 Fuchs-Frobenius theory 51 7 The classical formal theory of regular singular points 52 7.1 The exponential formalism xA 53 7.2 Non-resonance 54 7.3 Indicial polynomials 55 7.4 Regularity of differential systems 57 7.5 Regularity criterion for differential equations 58 7.6 Exponents 60 8 Jordan decomposition of differential modules 61 8.1 Jordan theory for differential modules 61 8.2 Action of commuting derivations 68 8.3 The regular case 69 8.4 Variant with parameters 72 9 Formal integrable connections (several variables) 76 9.1 Outline of Gérard-Levelt theory 76 9.2 Regularity and logarithmic extensions 80 Chapter IV Regularity: geometric theory 82 Introduction 82 10 Regularity and exponents along prime divisors 83 10.1 Transversal derivations and integral curves 84 10.2 Regular connections along prime divisors 89 10.3 Exponents along prime divisors 91 11 Regularity and exponents along a normal crossing divisor 92 11.1 Connections with logarithmic poles, and residues 92 11.2 Extensions with logarithmic poles 93 11.3 On reflexivity 94 11.4 Construction (and uniqueness) of 94 11.5 Local freeness of M 95 12 Base change 96 12.1 Restriction to curves I. The case when C meets D transversally at a smooth point 96 12.2 Restriction to curves II. The case when D is a strict normal crossing divisor 98 12.3 Restriction to curves III. The general case 101 12.4 Pull-back of a regular connection along D 105 13 Global regularity and exponents 105 13.1 Global regularity 105 13.2 Global exponents 107 Chapter V Irregularity: formal theory 108 Introduction 108 14 Confluent hypergeometric equations and phenomena related to irregularity 108 14.1 Solutions of the confluent hypergeometric equation 108 14.2 Meromorphic coefficients and Stokes multipliers 110 15 Poincaré rank 110 15.1 Spectral norms 110 15.2 Christol-Dwork-Katz theorem 116 15.3 Poincaré rank 118 16 Turrittin-Levelt decomposition and variants 120 16.1 The Turrittin-Levelt decomposition 120 16.2 Proof of the decomposition 122 17 Slopes and Newton polygons 125 17.1 Slope decomposition 125 17.2 Newton polygons 127 17.3 Newton polygons of cyclic modules 128 17.4 Index of operators and Malgrange's definition of irregularity 130 17.5 Variant with parameters. Turning points 131 17.6 Variation of the Newton polygon 133 18 Varia 137 18.1 Cyclic vectors in the neighborhood of a non-turning singular point 137 18.2 Turrittin decomposition around crossing points of the polar divisor 138 Chapter VI Irregularity: geometric theory 143 Introduction 143 19. Poincaré rank and Newton polygon (prime divisor). 143 19.1 Poincaré rank along a prime divisor 143 19.2 Newton polygon along a prime divisor 145 Stratificat19.3 ion of the polar divisor by Newton polygons 146 20 Turrittin-Levelt decomposition and -extensions 148 20.1 Formal Turrittin decomposition along a divisor 148 20.2 -extensions of irregular connections 149 21 Main theorem on the Poincaré rank 151 21.1 Statement of the main theorem 151 21.2 Proof of the main theorem 152 Chapter VII de Rham cohomology and Gauss-Manin connection 156 Introduction 156 22 Hypergeometric equation and Euler representation 156 23 de Rham cohomology and the Gauss-Manin connection 158 23.1 Direct image and higher direct images 158 23.2 de Rham and Spencer complexes 159 23.3 Some spectral sequences 162 23.4 Local construction of the Gauss-Manin connection 164 23.4 Flat base change 165 23.6 Vanishing and computation 166 24 Index formula 167 24.1 Deligne's global index formula on algebraic curves 167 24.2 Proof of the global index formula 168 Chapter VIII Elementary fibrations and applications 170 Introduction 170 25 Elementary fibrations and dévissage 171 25.1 Elementary fibrations 171 25.2 Artin sets 176 25.3 Dévissage 177 26 Main theorems on the Gauss-Manin connection 184 26.1 Generic finiteness of direct images 184 26.2 Generic base change for direct images 189 27 Gauss-Manin connection in the regular case 193 27.1 Main theorems (in the regular case) 193 27.2 Coherence of the cokernel of a regular connection 194 27.3 Regularity and exponents of the cokernel of a regular connection 199 Chapter IX Complex and p-adic comparison theorems 202 Introduction 202 28 The hypergeometric situation 202 29 Analytic contexts 203 29.1 Complex-analytic connections 203 29.2 Rigid analytic connections 206 30 Abstract comparison criteria 206 30.1 First criterion 207 30.2 Second criterion 208 31 Comparison theorem for algebraic vs. complex-analytic cohomology 209 31.1 Statement of the comparison 209 31.2 Reduction to the case of a rational elementary fibration 210 31.3 First way: reduction to an ordinary linear differential system 210 31.4 Second way: dealing with the relative situation 212 31.5 Deligne's GAGA version of the index formula 215 32 Comparison theorem for algebraic vs. rigid-analytic (regular coefficients) 216 32.1 Liouville numbers 216 32.2 Comparison 217 33 Rigid-analytic comparison theorem in relative dimension one 218 33.1 On the coherence of the cokernel of a connection in the rigid analytic situation 218 33.2 Rigid analytic comparison theorem in relative dimension one 221 34 Comparison theorem for algebraic vs. rigid-analytic (irregular coefficients) 226 34.1 Statement 226 34.2 Key propositions 226 34.3 Proof 227 34.4 Proof of 34.2.1 228 34.5 Proof of 34.2.2 234 34.6 Properties of the GAGA functor 235 Appendix A Riemann's existence theorem" in higher dimension, an elementary approach 237 Bibliography 241 Index 248 This is the revised second edition of the well-received book by the first two authors. It offers a systematic treatment of the theory of vector bundles with integrable connection on smooth algebraic varieties over a field of characteristic 0. Special attention is paid to singularities along divisors at infinity, and to the corresponding distinction between regular and irregular singularities. The topic is first discussed in detail in dimension 1, with a wealth of examples, and then in higher dimension using the method of restriction to transversal curves. The authors develop a new approach to classical algebraic/analytic comparison theorems in De Rham cohomology, and provide a unified discussion of the complex and the p-adic situations while avoiding the resolution of singularities. They conclude with a proof of a conjecture by Baldassarri to the effect that algebraic and p-adic analytic De Rham cohomologies coincide, under an arithmetic condition on exponents. As used in this text, the term “De Rham cohomology” refers to the hypercohomology of the De Rham complex of a connection with respect to a smooth morphism of algebraic varieties, equipped with the Gauss-Manin connection. This simplified approach suffices to establish the stability of crucial properties of connections based on higher direct images. The main technical tools used include: Artin local decomposition of a smooth morphism in towers of elementary fibrations, and spectral sequences associated with affine coverings and with composite functors.