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

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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