De Rham Cohomology of Differential Modules on Algebraic Varieties:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
2001
|
| Schriftenreihe: | Progress in Mathematics
189 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | 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) |
| Beschreibung: | 1 Online-Ressource (VII, 214 p) |
| ISBN: | 9783034883368 9783034895224 |
| DOI: | 10.1007/978-3-0348-8336-8 |
Internformat
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| 490 | 0 | |a Progress in Mathematics |v 189 | |
| 500 | |a 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) | ||
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Datensatz im Suchindex
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| author | André, Yves 1959- |
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| dewey-ones | 516 - Geometry |
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| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8336-8 |
| format | Electronic eBook |
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| isbn | 9783034883368 9783034895224 |
| language | English |
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| spelling | André, Yves 1959- Verfasser (DE-588)142060895 aut De Rham Cohomology of Differential Modules on Algebraic Varieties by Yves André, Francesco Baldassarri Basel Birkhäuser Basel 2001 1 Online-Ressource (VII, 214 p) txt rdacontent c rdamedia cr rdacarrier Progress in Mathematics 189 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) Mathematics Geometry Mathematik Differentialmodul (DE-588)4397443-0 gnd rswk-swf DeRham-Komplex (DE-588)4617507-6 gnd rswk-swf Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf DeRham-Komplex (DE-588)4617507-6 s Differentialmodul (DE-588)4397443-0 s Algebraische Varietät (DE-588)4581715-7 s 1\p DE-604 Baldassarri, Francesco 1951- Sonstige (DE-588)1077542771 oth https://doi.org/10.1007/978-3-0348-8336-8 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | André, Yves 1959- De Rham Cohomology of Differential Modules on Algebraic Varieties Mathematics Geometry Mathematik Differentialmodul (DE-588)4397443-0 gnd DeRham-Komplex (DE-588)4617507-6 gnd Algebraische Varietät (DE-588)4581715-7 gnd |
| subject_GND | (DE-588)4397443-0 (DE-588)4617507-6 (DE-588)4581715-7 |
| title | De Rham Cohomology of Differential Modules on Algebraic Varieties |
| title_auth | De Rham Cohomology of Differential Modules on Algebraic Varieties |
| title_exact_search | De Rham Cohomology of Differential Modules on Algebraic Varieties |
| title_full | De Rham Cohomology of Differential Modules on Algebraic Varieties by Yves André, Francesco Baldassarri |
| title_fullStr | De Rham Cohomology of Differential Modules on Algebraic Varieties by Yves André, Francesco Baldassarri |
| title_full_unstemmed | De Rham Cohomology of Differential Modules on Algebraic Varieties by Yves André, Francesco Baldassarri |
| title_short | De Rham Cohomology of Differential Modules on Algebraic Varieties |
| title_sort | de rham cohomology of differential modules on algebraic varieties |
| topic | Mathematics Geometry Mathematik Differentialmodul (DE-588)4397443-0 gnd DeRham-Komplex (DE-588)4617507-6 gnd Algebraische Varietät (DE-588)4581715-7 gnd |
| topic_facet | Mathematics Geometry Mathematik Differentialmodul DeRham-Komplex Algebraische Varietät |
| url | https://doi.org/10.1007/978-3-0348-8336-8 |
| work_keys_str_mv | AT andreyves derhamcohomologyofdifferentialmodulesonalgebraicvarieties AT baldassarrifrancesco derhamcohomologyofdifferentialmodulesonalgebraicvarieties |