Author	Andrรฉ, Yves. author
Title	De Rham Cohomology of Differential Modules on Algebraic Varieties [electronic resource] / by Yves Andrรฉ, Francesco Baldassarri
Imprint	Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2001
Connect to	http://dx.doi.org/10.1007/978-3-0348-8336-8
Descript	VII, 214 p. online resource

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 differยญ 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 coeffiยญ 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)

1 Regularity in several variables -- ยง1 Geometric models of divisorially valued function fields -- ยง2 Logarithmic differential operators -- ยง3 Connections regular along a divisor -- ยง4 Extensions with logarithmic poles -- ยง5 Regular connections: the global case -- ยง6 Exponents -- Appendix A: A letter of Ph. Robba (Nov. 2, 1984) -- Appendix B: Models and log schemes -- 2 Irregularity in several variables -- ยง1 Spectral norms -- ยง2 The generalized Poincarรฉ-Katz rank of irregularity -- ยง3 Some consequences of the Turrittin-Levelt-Hukuhara theorem -- ยง4 Newton polygons -- ยง5 Stratification of the singular locus by Newton polygons -- ยง6 Formal decomposition of an integrable connection at a singular divisor -- ยง7 Cyclic vectors, indicial polynomials and tubular neighborhoods -- 3 Direct images (the Gauss-Manin connection) -- ยง1 Elementary fibrations -- ยง2 Review of connections and De Rham cohomology -- ยง3 Dรฉvissage -- ยง4 Generic finiteness of direct images -- ยง5 Generic base change for direct images -- ยง6 Coherence of the cokernel of a regular connection -- ยง7 Regularity and exponents of the cokernel of a regular connection -- ยง8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case) -- Appendix C: Berthelotโ{128}{153}s comparison theorem on OXDX-linear duals -- Appendix D: Introduction to Dworkโ{128}{153}s algebraic dual theory -- 4 Complex and p-adic comparison theorems -- ยง1 Review of analytic connections and De Rham cohomology -- ยง2 Abstract comparison criteria -- ยง3 Comparison theorem for algebraic vs.complex-analytic cohomology -- ยง4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients) -- ยง5 Rigid-analytic comparison theorem in relative dimension one -- ยง6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients) -- ยง7 The relative non-archimedean Turrittin theorem -- Appendix E: Riemannโ{128}{153}s โ{128}{156}existence theoremโ{128}{157} in higher dimension, an elementary approach -- References