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De Rham Cohomology of Differential Modules on Algebraic Varieties

Part of the Progress in Mathematics Book Series (volume 189)

DOI: 10.1007/978-3-030-39719-7

ISBN: 978-3-030-39718-0

Publisher: Springer International Publishing - Springer Journals

Jul 17, 2020

1 - 9 of 9 Chapters

LitStream

Chapter 1Differential algebra

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 1 –22

[We start by introducing the Gauss and Kummer hypergeometric differential equations. These classical equations will reenter the stage at the beginning of several later chapters, serving as guiding examples to motivate and illustrate their content.]
LitStream

Chapter 2Connections on algebraic varieties

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 23 –33

[Most of this book deals with differential systems in a geometric context, in several variables. In order to give such systems an intrinsic meaning, two languages are at disposal, which are in principle equivalent: integrable connections and D-modules.]
LitStream

Chapter 3Regularity: formal theory

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 35 –68

[With the example of Gauss hypergeometric equation at hand, we first recall the classical Fuchs-Frobenius theory of meromorphic linear differential systems: Fuchs’ numerical criterion of regularity and Frobenius’ convergence of the asymptotic expansions of the solutions.]
LitStream

Chapter 4Regularity: geometric theory

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 69 –94

[The central topic of this chapter is the notion of regularity in several variables.The aim of this chapter is to provide a purely algebraic, systematic treatment of these questions (avoiding resolution of singularities beyond the case of surfaces).]
LitStream

Chapter 5Irregularity: formal theory

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 95 –129

[With the example of Kummer hypergeometric equation at hand, we first recall some typical issues due to an irregular singularity: presence of divergent series, tension between the topological viewpoint (monodromy) and the formal algebraic viewpoint and its resolution in terms of asymptotic...
LitStream

Chapter 6Irregularity: geometric theory

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 131 –143

[This chapter completes our study of irregularity in several variables, in the geometric framework of integrable connections.]
LitStream

Chapter 7de Rham cohomology and Gauss-Manin connection

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 145 –158

[This chapter opens the cohomological part of the book.]
LitStream

Chapter 8Elementary fibrations and applications to Gauss-Manin

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 159 –190

[This chapter develops a new approach to the study of direct images of connections (i.e., de Rham cohomology with coefficients, endowed with the Gauss-Manin connection) with respect to a smooth, not necessarily proper, morphism of smooth algebraic varieties in characteristic 0.]
LitStream

Chapter 9Complex and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \mathit p$$ \end{document}-adic comparison theorems

André, Yves; Baldassarri, Francesco; Cailotto, Maurizio

Pages: 191 –225

[In this chapter, we show how to adapt Artin’s strategy of proof of his comparison theorem for ´etale cohomology [2] in the de Rham context.]

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