De Rham cohomology of differential modules on algebraic varieties

by André, Yves

Type Book

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Title Statements

Full title :De Rham cohomology of differential modules on algebraic varieties / Yves André, Francesco Baldassarri, Maurizio Cailotto.

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

Edition Statements

Publication Date:2020.
Edition:2nd ed

Publication Statements

Publisher(s):Birkhäuser
Place of Publication:Cham
Language:eng
Publication Date:2020.

Notes

14 Confluent hypergeometric equations and phenomena related to irregularity

The Physical Object

Description:1 online resource (250 pages)

Edition Identifiers

ISBN:9783030397197, 303039719X
DOI:10.1007/978-3-030-39719-7
oclc:on1181835860
LC Classification:QA612.3
Dewey Decimal:514/.23

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

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 -- Introdu…