De Rham Cohomology of Differential Modules on Algebraic Varieties
2nd Edition
Published on 17. July 2021
Book
Paperback/Softback
XIV, 241 pages
978-3-030-39721-0 (ISBN)
Description
This is a study of algebraic differential modules in several variables, and of some of their relations with analytic differential modules. Let us explain its source. The idea of computing the cohomology of a manifold, in particular its Betti numbers, by means of differential forms goes back to E. Cartan and G. De Rham. In the case of a smooth complex algebraic variety X, there are three variants: i) using the De Rham complex of algebraic differential forms on X, ii) using the De Rham complex of holomorphic differential forms on the analytic an manifold X underlying X, iii) using the De Rham complex of Coo complex differential forms on the differ entiable manifold Xdlf underlying Xan. These variants tum out to be equivalent. Namely, one has canonical isomorphisms of hypercohomology: While the second isomorphism is a simple sheaf-theoretic consequence of the Poincare lemma, which identifies both vector spaces with the complex cohomology H (XtoP, C) of the topological space underlying X, the first isomorphism is a deeper result of A. Grothendieck, which shows in particular that the Betti numbers can be computed algebraically. This result has been generalized by P. Deligne to the case of nonconstant coeffi cients: for any algebraic vector bundle .M on X endowed with an integrable regular connection, one has canonical isomorphisms The notion of regular connection is a higher dimensional generalization of the classical notion of fuchsian differential equations (only regular singularities).
Reviews / Votes
"If I had to summarize the differences in the exposition in one sentence, I would say that the authors manage to make the transition from a research monograph to an exposition that reads more like an advanced textbook, while retaining the rigor and the scientific interest; it is an example of what may be called a 'research textbook'." (?Adolfo Quiros, Mathematical Reviews, January, 2023)"If I had to summarize the differences in the exposition in one sentence, I would say that the authors manage to make the transition from a research monograph to an exposition that reads more like an advanced textbook, while retaining the rigor and the scientific interest; it is an example of what may be called a 'research textbook'." (?Adolfo Quiros, Mathematical Reviews, January, 2023)
More details
Product info
Paperback
Series
Edition
2nd ed. 2020
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Professional and scholarly
Edition type
Revised edition
Illustrations
XIV, 241 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 15 mm
Weight
394 gr
ISBN-13
978-3-030-39721-0 (9783030397210)
DOI
10.1007/978-3-030-39719-7
Schweitzer Classification
Other editions
Additional editions
Yves André | Francesco Baldassarri | Maurizio Cailotto
De Rham Cohomology of Differential Modules on Algebraic Varieties
Book
07/2020
2nd Edition
Birkhäuser
€128.39
Shipment within 7-9 days
Content
1 Regularity in several variables.- ?1 Geometric models of divisorially valued function fields.- ?2 Logarithmic differential operators.- ?3 Connections regular along a divisor.- ?4 Extensions with logarithmic poles.- ?5 Regular connections: the global case.- ?6 Exponents.- Appendix A: A letter of Ph. Robba (Nov. 2, 1984).- Appendix B: Models and log schemes.- 2 Irregularity in several variables.- ?1 Spectral norms.- ?2 The generalized Poincare-Katz rank of irregularity.- ?3 Some consequences of the Turrittin-Levelt-Hukuhara theorem.- ?4 Newton polygons.- ?5 Stratification of the singular locus by Newton polygons.- ?6 Formal decomposition of an integrable connection at a singular divisor.- ?7 Cyclic vectors, indicial polynomials and tubular neighborhoods.- 3 Direct images (the Gauss-Manin connection).- ?1 Elementary fibrations.- ?2 Review of connections and De Rham cohomology.- ?3 Devissage.- ?4 Generic finiteness of direct images.- ?5 Generic base change for direct images.- ?6 Coherence of the cokernel of a regular connection.- ?7 Regularity and exponents of the cokernel of a regular connection.- ?8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case).- Appendix C: Berthelot's comparison theorem on OXDX-linear duals.- Appendix D: Introduction to Dwork's algebraic dual theory.- 4 Complex and p-adic comparison theorems.- ?1 Review of analytic connections and De Rham cohomology.- ?2 Abstract comparison criteria.- ?3 Comparison theorem for algebraic vs.complex-analytic cohomology.- ?4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients).- ?5 Rigid-analytic comparison theorem in relative dimension one.- ?6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients).- ?7 The relative non-archimedean Turrittin theorem.- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach.- References.