Walther, Uli Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements. (English) Zbl 1070.32021 Compos. Math. 141, No. 1, 121-145 (2005). By definition, a generic central hyperplane arrangement in \({\mathbb C}^n\), \(n \geq 2,\) is the union of \(k\) hyperplanes given by linear forms \(h_i\), \(i = 1, \ldots, k,\) such that each subset of \(\min \{k,n\}\) of the hyperplanes cuts out the origin; such arrangement is defined by the product \(f = h_1 \cdots h_k.\) The aim of the paper under review is to compute the Bernstein-Sato polynomial \(b_f(s).\)First the author proves that \(b_f(s)\) divides the polynomial \(P(s) = (s+1)^{n-1}\prod_{i=0}^{2k-n-2}(s+{i+n \over k}).\) Based on properties of the relative de Rham complex [B. Malgrange, Semin. Goulaouic-Schwartz 1973–1974, Equat. dériv. part. Analyse fonct., Exposé XX (1974; Zbl 0308.32008)] and D-module algorithm for computing integration functors [T. Oaku and N. Takayama, J. Pure Appl. Algebra 156, No. 2–3, 267–308 (2001; Zbl 0983.13008)] he then shows that the degrees of differential forms corresponding to generators for the top cohomology group of the Milnor fiber associated with a homogeneous polynomial \(Q\) of degree \(k>0\) are roots of \(b_Q(s).\) Furthermore, from the obtained results it follows that the Bernstein-Sato polynomial of a generic central arrangement of \(k\) hyperplanes is equal to \(P(s)\) or \(P(s)/(s+1).\)Finally, the author remarks that M. Saito points out [Multiplier ideals, b-functions, and spectrum, Preprint, 2004, arXiv:math.AG/0402363] that the Hodge theory can be used to conclude that only the first possibility can be realized. In addition, some cases of a conjecture of P. Orlik and R. Randell [Ark. Mat. 31, No. 1, 71–81 (1993; Zbl 0807.32029)] on the cohomology of the Milnor fiber of a generic central arrangement are verified. In the case of non-generic arrangement an interesting idea on connections between the roots of \(b_f(s)\) and certain subschemes of the arrangement (called “iterated singular loci”) is discussed. In conclusion, the author also considers some related questions and conjectures with interpretation in the context of the theory of logarithmic differential forms and vector fields [F. J. Calderón-Moreno, Ann. Sci. Ec. Norm. Super., IV. Ser. 32, No. 5, 701–714 (1999; Zbl 0955.14013)]. Reviewer: Aleksandr G. Aleksandrov (Moskva) Cited in 21 Documents MSC: 32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) 32S22 Relations with arrangements of hyperplanes 14F40 de Rham cohomology and algebraic geometry 13N10 Commutative rings of differential operators and their modules Keywords:Bernstein-Sato polynomial; Milnor fibers; relative de Rham complex; de Rham cohomology; hyperplane arrangements; holonomic D-modules; integration functors; logarithmic differential forms Citations:Zbl 0308.32008; Zbl 0983.13008; Zbl 0807.32029; Zbl 0955.14013 Software:Macaulay2; Kan PDFBibTeX XMLCite \textit{U. Walther}, Compos. Math. 141, No. 1, 121--145 (2005; Zbl 1070.32021) Full Text: DOI arXiv