De Rham Cohomology of Differential Modules on Algebraic VarietiesThis 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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Contents
Regularity in several variables | 1 |
4 Comparison theorem for algebraic vs rigidanalytic | 4 |
Irregularity in several variables | 49 |
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De Rham Cohomology of Differential Modules on Algebraic Varieties Yves André,Francesco Baldassarri No preview available - 2012 |
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algebraic assume base change bicomplex canonical closed immersion closed point codimension Coh(X coherent Ox-module coherent sheaf coordinatized elementary fibration Corollary cyclic vector D₁ decomposition defined denote Diff diff.op differential module differential operators direct image divisor divisorially valued function Dx/s étale covering étale morphism étale topology exact sequence exists exponents extension F/K-differential module fiber finite rank follows formal function field functor Gauss-Manin connection hence indicial polynomial induces integrable connection inverse image isomorphism Lemma Let Ɛ Let f locally closed locally free log dominant MIC(X morphism of smooth Newton polygon open subset Poincaré-Katz rank Proof Proposition rational elementary fibration RDR f*(E replace resp Rham complex sheaf with integrable simple zero smooth connected smooth K-model smooth K-varieties smooth morphism spectral sequence theorem transversal tubular neighborhood valuation valued function field