Bernstein-Sato ideals and local systems
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, p. 549-603

The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of Malgrange and Kashiwara which states that the Bernstein-Sato polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor fibers of the hypersurface. We also address a multi-variable version of the Monodromy Conjecture, prove that it follows from the usual single-variable Monodromy Conjecture, and prove it in the case of hyperplane arrangements.

La topologie des variétés lisses quasi-projectives est très restrictive. Par exemple, les lieux des sauts pour la cohomologie des systèmes locaux sont des translatés par un point de torsion des sous-tores d’un tore complexe. Nous proposons et confirmons partiallement une relation entre idéaux de Bernstein-Sato et systèmes locaux. Cela donne une nouvelle perspective sur la structure des lieux des sauts pour la cohomologie. Le résultat principal est une généralisation partielle pour le cas de plusieurs polynômes du théorème de Malgrange et Kashiwara qui affirme que le polynôme de Bernstein-Sato d’une hypersurface donne les valeurs propres de la monodromie sur les fibres du Milnor de l’hypersurface. Nous abordons aussi une version à plusieurs variables de la Conjecture de Monodromie, nous prouvons qu’elle résulte de la version à une variable, et nous la prouvons pour les arrangements d’hyperplanes.

DOI : https://doi.org/10.5802/aif.2939
Classification:  14F10,  32S40,  14B05,  32S05,  32S22
Keywords: Bernstein-Sato ideal, Bernstein-Sato polynomial, b-function, 𝒟-modules, local systems, cohomology jump loci, characteristic variety, Sabbah specialization, Alexander module, Milnor fiber, Monodromy Conjecture, hyperplane arrangements.
@article{AIF_2015__65_2_549_0,
     author = {Budur, Nero},
     title = {Bernstein-Sato ideals and local systems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     pages = {549-603},
     doi = {10.5802/aif.2939},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_2_549_0}
}
Budur, Nero. Bernstein-Sato ideals and local systems. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 549-603. doi : 10.5802/aif.2939. http://www.numdam.org/item/AIF_2015__65_2_549_0/

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