By Yves André

ISBN-10: 3034883366

ISBN-13: 9783034883368

ISBN-10: 3034895224

ISBN-13: 9783034895224

This is a research of algebraic differential modules in different variables, and of a few in their kinfolk with analytic differential modules. allow us to clarify its resource. the belief of computing the cohomology of a manifold, particularly its Betti numbers, through differential varieties is going again to E. Cartan and G. De Rham. with regards to a tender advanced algebraic kind X, there are 3 versions: i) utilizing the De Rham complicated of algebraic differential varieties on X, ii) utilizing the De Rham complicated of holomorphic differential kinds at the analytic an manifold X underlying X, iii) utilizing the De Rham complicated of Coo complicated differential types at the vary entiable manifold Xdlf underlying Xan. those versions tum out to be similar. specifically, one has canonical isomorphisms of hypercohomology: whereas the second one isomorphism is a straightforward sheaf-theoretic end result of the Poincare lemma, which identifies either vector areas with the advanced cohomology H (XtoP, C) of the topological area underlying X, the 1st isomorphism is a deeper results of A. Grothendieck, which exhibits particularly that the Betti numbers should be computed algebraically. This outcome has been generalized via P. Deligne to the case of nonconstant coeffi cients: for any algebraic vector package .M on X endowed with an integrable commonplace connection, one has canonical isomorphisms The concept of standard connection is the next dimensional generalization of the classical inspiration of fuchsian differential equations (only typical singularities).

**Read or Download De Rham Cohomology of Differential Modules on Algebraic Varieties PDF**

**Best algebraic geometry books**

This ebook provides an creation to trendy geometry. ranging from an effortless point the writer develops deep geometrical suggestions, taking part in a big position these days in modern theoretical physics. He offers a variety of suggestions and viewpoints, thereby displaying the relatives among the choice ways.

Surveys, tutorials, and learn papers from a summer season 2002 workshop research a number themes in algebraic geometry and geometric modeling. Papers are divided into sections on modeling curves and surfaces, multisided patches, implicitization and parametrization, subject forms, and combined quantity and resultants, and papers from either disciplines are incorporated in every one part.

**Download e-book for kindle: Brauer groups, Tamagawa measures, and rational points on by Jorg Jahnel**

The significant subject matter of this publication is the examine of rational issues on algebraic different types of Fano and intermediate type--both by way of while such issues exist and, in the event that they do, their quantitative density. The ebook involves 3 components. within the first half, the writer discusses the idea that of a top and formulates Manin's conjecture at the asymptotics of rational issues on Fano forms.

- Measure, Topology, and Fractal Geometry
- Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 3 Folge)
- Heegner Modules and Elliptic Curves
- Intersections de deux quadriques et pinceaux de courbes de genre 1: Intersections of two quadrics and pencils of curves of genus 1
- Elliptic Curves: Notes from Postgraduate Lectures Given in Lausanne 1971/72

**Extra resources for De Rham Cohomology of Differential Modules on Algebraic Varieties**

**Example text**

Yn). Soit M un K(x, y)-module differentiel integrable libre de rang s. Consicterons Y comme des parametres et M comme un K(y)(x)-module differentiel. Alors Theoreme: Soit a independants de y. E K. Les exposants de ce module differentiel en a sont Demonstration. Prenons a = O. Soil F = (::) une b",e de M. Les motrires de derivation sont o -F=HF ox ou H, G i E o -F=Gi F °Yi ' Mat (s, K (x, y)). L' hypothese que M est integrable se traduit par o 0 - H +HG' = -G· +G ·H 0Yi I ox I I pour I :s i :s n.

O (ei is the valuation of the image in Oc,Q of a local equation of Zi in Ox"p). 37 Regularity in several variables Now V induces a composite map The map Ui is nothing but the specialization of Reszj V at P, and v 0 U is nothing but ResQh*V. Since the Reszj V mutually commute, the endomorphisms v 0 Ui = eiUi of the finite dimensional K = K{Q)-vector space (h*£)Q ® K{Q) also commute and the eigenvalues of Li v 0 Ui are of the form Li eiAi, for eigenvalues Ai of Ui. We conclude that EXPQ{h*V) C (L ei Expzj (V)) + z.

1 The exponents (at v) of a regular F I K -differential module £ are the exponents of the induced FIC-differential module (in one variable x) £IC. 3). 2 Expv(£) C K. 2). 4 Exponents of a regular connection along a divisor Let (X, Z) be a model of a divisorially valued function field (F, v). 1». 1 The exponents of V along Z are the exponents of Expz(V) = EXPv(£). £ at v. 2 Let (X, Z)beamodeland(E, V), (EI, V]), (E2, V2)becoherent sheaves with integrable regular connections on X \ Z. Regularity in several variables 35 i) If £ sits in a horizontal exact sequence 0 ----+ £) ----+ £ ----+ £2 ----+ 0, then Expz(V) = Expz(V]) U EXpZ(V2).

### De Rham Cohomology of Differential Modules on Algebraic Varieties by Yves André

by Robert

4.5