×

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
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] Borel, A. : Algebraic D-Modules , Academic Press, Boston, 1987. · Zbl 0642.32001
[2] Deligne, P. : Equation différentielle à points singuliers réguliers , Lecture Notes in Math. 163, Springer-Verlag, Berlin, 1970. · Zbl 0244.14004
[3] Dimca, A. : On the Milnor fibration of weighted homogeneous polynomials , Compositio Math. 76 (1990), 19-47. · Zbl 0726.14002
[4] Grothendieck, A. : On the de Rham cohomology of algebraic varieties , Publ. Math. IHES 29 (1966), 95-103. · Zbl 0145.17602
[5] Grothendieck, A. and Dieudonné, J. : Eléments de géométrie algébrique IV , Publ. Math. IHES 32 (1967). · Zbl 0153.22301
[6] Kashiwara, M. : B-function and holonomic systems, Inv. Math. 38 (1976), 33-53. · Zbl 0354.35082
[7] Pham, F. : Singularité des systèmes differéntiels de Gauss-Manin , Progress in Math. 2,, Birkhäuser, 1979. · Zbl 0524.32015
[8] Saito, M. : On the structure of Brieskorn lattice , Ann. Institut Fourier 39 (1989), 27-72. · Zbl 0644.32005
[9] Saito, M. : Induced D-modules and differential complexes , Bull. Soc. Math. France 117 (1989), 361-387. · Zbl 0705.32005
[10] Sato, M. , Kawai, T. and Kashiwara, M. : Microfunctions and pseudodifferential equations , in Lecture Notes in Math. vol. 287, Springer, Berlin (1973), 264-529. · Zbl 0277.46039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.