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La classe fondamentale d’une variété algébrique engendre le \({\mathcal D}\)-module qui calcule sa cohomologie d’intersection. (D’après Masaki Kashiwara). (French) Zbl 0568.14010
Systèmes différentielles et singularités, Colloq. Luminy/France 1983, Astérisque 130, 260-271 (1985).
[For the entire collection see Zbl 0559.00004.]
Let X be a complex algebraic variety, \({\mathcal D}_ X\) the sheaf of differential operators on X, and Y a hypersurface defined by \(F(x)=0\). Put \({\mathcal L}(Y,X)\) the \({\mathcal D}_ X\)-submodule in \({\mathcal H}^ 1_ Y({\mathcal O})\) corresponding to the intersection complex \(i_*IC_ Y^{\bullet}\) by the Riemann-Hilbert correspondence DR. If \(\xi\) is a vector field on X such that \(\xi\) (F) does not vanish on Y, then the cohomology class [\(\xi\) (F)/F] in \(H_ Y^ 1({\mathcal O}_ X)\) is a generator of \({\mathcal L}(Y,X)\). This is the first version of a theorem obtained by Kashiwara (in a letter dated 16 May 1983 from Kashiwara to the author). In this paper, the author reports on the generalized version of the theorem to the case that Y is an algebraic subvariety of X of codimension d. Under some suitable conditions, Kashiwara defined \(C_{Y| X}\) as a fundamental class of \(H^ d_ Y(X,\Omega^ d_ X)\), and proved that \(\Omega^ d_ X\otimes_{{\mathcal O}_ X}{\mathcal L}(Y,X)\subset \Omega^ d_ X\otimes_{{\mathcal O}_ X}{\mathcal H}^ d_ Y({\mathcal O}_ X)={\mathcal H}^ d_ X(\Omega^ d_ X)\). As a corollary, it is proved that \(\xi C_{Y| X}\) generates \({\mathcal L}(,Y)\) as a \({\mathcal D}_ X\)-module.
Reviewer: M.Muro

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
58J10 Differential complexes
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
32K15 Differentiable functions on analytic spaces, differentiable spaces
Zbl 0559.00004