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

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
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### References:

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