| Description |
1 online resource (250 pages) |
| Series |
Progress in Mathematics ; v. 189 |
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Progress in mathematics (Boston, Mass.) ; v. 189.
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| Contents |
Intro -- 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 |
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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 |
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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 |
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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 |
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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 |
| Summary |
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 |
| Notes |
14 Confluent hypergeometric equations and phenomena related to irregularity |
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Print version record |
| Subject |
Homology theory.
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Differential algebra.
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Modules (Algebra)
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Homología, Teoría de
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Álgebra diferencial
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Differential algebra
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Homology theory
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Modules (Algebra)
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| Form |
Electronic book
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| Author |
Baldassarri, F. (Francesco), 1951-
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Cailotto, Maurizio
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| ISBN |
9783030397197 |
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303039719X |
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