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On twisted de Rham cohomology. (English) Zbl 0915.14012

The paper deals with the twisted de Rham complex of global differential forms on the complement of the divisor \(f_1 \cdots f_r = 0\) in the affine space \({\mathbb A}^n\) endowed with the differential \(d + (dg - \sum_{i=1}^r \beta_i df_i/f_i) \wedge,\) where \(d\) is the usual exterior derivative, \(g, f_1, \ldots, f_r\) are polynomials, and \(\beta_1, \ldots, \beta_r\) are complex numbers. Using the Laplace transform [B.Dwork, “Generalized hypergeometric functions” (1990; Zbl 0747.33001)] and their previous results [see e.g. A. Adolphson and S. Sperber, J. Reine Angew. Math. 443, 151-177 (1993; Zbl 0853.11067)] the authors compute the cohomology of this and some related complexes for generic \(g, f_1, \ldots, f_r, \beta_1, \ldots, \beta_r.\)

MSC:

14F40 de Rham cohomology and algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
33C20 Generalized hypergeometric series, \({}_pF_q\)
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