On the cohomology of a general fiber of a polynomial map. (English) Zbl 0824.14016

Dimca, Alexandru; Saito, Morihiko

Compos. Math. 85, No. 3, 299-309 (1993).

Let \(F\) be the generic fiber of a polynomial function \(f : \mathbb{C}^ n \to \mathbb{C}\). Let \(\Omega^ \bullet\) denote the complex of global algebraic differential forms on \(\mathbb{C}^ n\). Define a differential \(D_ f\) on \(\Omega^ \bullet\) by \(D_ f (\omega) = d \omega - df \wedge \omega\). Then one has an isomorphism \(H^{k + 1} (\Omega^ \bullet, D_ f) \simeq \widetilde H^ k (F; \mathbb{C})\) for any \(k \geq 0\). The proof uses the theory of algebraic Gauss-Manin systems and the theory of monodromic algebraic \({\mathcal D}\)-modules.

Reviewer: A.Dimca (Bordeaux)

MSC:

14F40	de Rham cohomology and algebraic geometry
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38	Sheaves of differential operators and their modules, \(D\)-modules
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

de Rham cohomology; \({\mathcal D}\)-modules; algebraic differential forms; Gauss-Manin systems

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