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De Rham Cohomology of Differential Modules on Algebraic Varieties

  • Yves André
  • Francesco Baldassarri

Part of the Progress in Mathematics book series (PM, volume 189)

Table of contents

  1. Front Matter
    Pages N1-vii
  2. Yves André, Francesco Baldassarri
    Pages 1-48
  3. Yves André, Francesco Baldassarri
    Pages 49-102
  4. Yves André, Francesco Baldassarri
    Pages 103-169
  5. Yves André, Francesco Baldassarri
    Pages 171-208
  6. Back Matter
    Pages 209-214

About this book

Introduction

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).

Keywords

Dimension Divisor Geometrie Grad algebra algebraic geometry algebraic varieties

Authors and affiliations

  • Yves André
    • 1
  • Francesco Baldassarri
    • 2
  1. 1.Institut de MathématiquesUniversité Pierre et Marie CurieParis Cedex 05
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversità degli Studi di PadovaItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-8336-8
  • Copyright Information Birkhäuser Verlag 2001
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-9522-4
  • Online ISBN 978-3-0348-8336-8
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site