De Rham Cohomology of Differential Modules on Algebraic Varieties

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).

List of contents

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 Poincaré-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 Dévissage.-
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-analyticcohomology (irregular coefficients).-
7 The relative non-archimedean Turrittin theorem.- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach.- References.

Summary

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).

Product details

• Authors: Yves Andre, Yve André, Yves André, Francesco Baldassari, Francesco Baldassarri
• Publisher: Birkhäuser
• Languages: English
• Product format: Hardback
• Released: 01.01.2007
• EAN: 9783764363482
• ISBN: 978-3-7643-6348-2
• No. of pages: 214
• Weight: 499 g
• Series: Progress in Mathematics; Progress in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; Algebra, B, geometry, Grad, Mathematics and Statistics, Algebraic Geometry, algebraic varieties