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.
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.
@article{AIF_2015__65_2_549_0,
author = {Budur, Nero},
title = {Bernstein-Sato ideals and local systems},
journal = {Annales de l'Institut Fourier},
volume = {65},
year = {2015},
pages = {549-603},
doi = {10.5802/aif.2939},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2015__65_2_549_0}
}
Budur, Nero. Bernstein-Sato ideals and local systems. Annales de l'Institut Fourier, Tome 65 (2015) pp. 549-603. doi : 10.5802/aif.2939. http://gdmltest.u-ga.fr/item/AIF_2015__65_2_549_0/
[1] Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves, Bull. Amer. Math. Soc. (N.S.), Tome 26 (1992) no. 2, pp. 310-314 | Article | MR 1129312 | Zbl 0759.14016
[2] Geometry of cohomology support loci for local systems. I, J. Algebraic Geom., Tome 6 (1997) no. 3, pp. 563-597 | MR 1487227 | Zbl 0923.14010
[3] Démonstration constructive de l’existence de polynômes de Bernstein-Sato pour plusieurs fonctions analytiques, Compos. Math., Tome 141 (2005) no. 1, pp. 175-191 | Article | MR 2099775 | Zbl 1099.32005
[4] Local Bernstein-Sato ideals: algorithm and examples, J. Symbolic Comput., Tome 45 (2010) no. 1, pp. 46-59 | Article | MR 2568898 | Zbl 1184.14030
[5] Analytic -modules and applications, Kluwer Academic Publishers Group, Dordrecht, Mathematics and its Applications, Tome 247 (1993), pp. xiv+581 | Article | MR 1232191 | Zbl 0805.32001
[6] Heights in Diophantine geometry, Cambridge University Press, Cambridge, New Mathematical Monographs, Tome 4 (2006), pp. xvi+652 | Article | MR 2216774 | Zbl 1115.11034
[7] Constructibilité de l’idéal de Bernstein, Singularities—Sapporo 1998, Kinokuniya, Tokyo (Adv. Stud. Pure Math.) Tome 29 (2000), pp. 79-95 | Zbl 1070.14507
[8] Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque (1986) no. 140-141, p. 3-134, 251 (Géométrie et analyse microlocales) | MR 864073 | Zbl 0624.32009
[9] Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers, Adv. Math., Tome 221 (2009) no. 1, pp. 217-250 | Article | MR 2509325 | Zbl 1187.14024
[10] The monodromy conjecture for hyperplane arrangements, Geom. Dedicata, Tome 153 (2011), pp. 131-137 | Article | MR 2819667 | Zbl 1227.32035
[11] Cohomology jump loci of quasi-projective varieties (http://arxiv.org/abs/1211.3766, to appear in Ann. Sci. École Norm. Sup.) | MR 3335842
[12] Multivariable Hodge theoretical invariants of germs of plane curves, J. Knot Theory Ramifications, Tome 20 (2011) no. 6, pp. 787-805 | Article | MR 2812263 | Zbl 1226.32016
[13] Singular 3-1-3 — A computer algebra system for polynomial computations (http://www.singular.uni-kl.de (Feb 2010)) | Zbl 0902.14040
[14] Singularities and topology of hypersurfaces, Springer-Verlag, New York, Universitext (1992), pp. xvi+263 | Article | MR 1194180 | Zbl 0753.57001
[15] Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc., Tome 359 (2007) no. 7, pp. 3505-3528 | Article | MR 2299465 | Zbl 1119.32012
[16] Nonabelian cohomology jump loci from an analytic viewpoint (http://arxiv.org/abs/1206.3773) | MR 3231055
[17] Topology and geometry of cohomology jump loci, Duke Math. J., Tome 148 (2009) no. 3, pp. 405-457 | Article | MR 2527322 | Zbl 1222.14035
[18] Lectures on -modules (1998) (www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf)
[19] Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math., Tome 90 (1987) no. 2, pp. 389-407 | Article | MR 910207 | Zbl 0659.14007
[20] Higher obstructions to deforming cohomology groups of line bundles, J. Amer. Math. Soc., Tome 4 (1991) no. 1, pp. 87-103 | Article | MR 1076513 | Zbl 0735.14004
[21] Bernstein-Sato’s polynomial for several analytic functions, J. Math. Kyoto Univ., Tome 33 (1993) no. 2, pp. 399-411 | MR 1231750 | Zbl 0797.32007
[22] Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982), Springer, Berlin (Lecture Notes in Math.) Tome 1016 (1983), pp. 134-142 | Article | MR 726425 | Zbl 0566.32022
[23] -functions and holonomic systems. Rationality of roots of -functions, Invent. Math., Tome 38 (1976/77) no. 1, pp. 33-53 | Article | MR 430304 | Zbl 0354.35082
[24] Équations diophantiennes exponentielles, Invent. Math., Tome 78 (1984) no. 2, pp. 299-327 | Article | MR 767195 | Zbl 0554.10009
[25] textttdmod.lib (A Singular 3-1-3 library for algebraic D-modules (2011))
[26] Eigenvalues for the monodromy of the Milnor fibers of arrangements, Trends in singularities, Birkhäuser, Basel (Trends Math.) (2002), pp. 141-150 | MR 1900784 | Zbl 1036.32019
[27] Non vanishing loci of Hodge numbers of local systems, Manuscripta Math., Tome 128 (2009) no. 1, pp. 1-31 | Article | MR 2470184 | Zbl 1160.14004
[28] Fonctions d’Igusa -adiques et polynômes de Bernstein, Amer. J. Math., Tome 110 (1988) no. 1, pp. 1-21 | Article | MR 926736 | Zbl 0644.12007
[29] Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants, Ann. Sci. École Norm. Sup. (4), Tome 22 (1989) no. 3, pp. 435-471 | Numdam | MR 1011989 | Zbl 0718.11061
[30] Polynômes de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 101 (1983), pp. 243-267 | MR 737934
[31] Generalized monodromy conjecture in dimension two, Geom. Topol., Tome 16 (2012) no. 1, pp. 155-217 | Article | MR 2872581 | Zbl 1241.14004
[32] Zeta functions and Alexander modules (http://arxiv.org/abs/math/0404212)
[33] An algorithm of computing -functions, Duke Math. J., Tome 87 (1997) no. 1, pp. 115-132 | Article | MR 1440065 | Zbl 0893.32009
[34] An algorithm for de Rham cohomology groups of the complement of an affine variety via -module computation, J. Pure Appl. Algebra, Tome 139 (1999) no. 1-3, pp. 201-233 (Effective methods in algebraic geometry (Saint-Malo, 1998)) | Article | MR 1700544 | Zbl 0960.14008
[35] Arrangements of hyperplanes, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 300 (1992), pp. xviii+325 | Article | MR 1217488 | Zbl 0757.55001
[36] Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. Lond. Math. Soc. (3), Tome 100 (2010) no. 3, pp. 795-834 | Article | MR 2640291 | Zbl 1273.55003
[37] Generic vanishing theory via mixed Hodge modules (http://arxiv.org/abs/1112.3058) | MR 3090229
[38] Holomorphy of Igusa’s and topological zeta functions for homogeneous polynomials, Pacific J. Math., Tome 201 (2001) no. 2, pp. 429-440 | Article | MR 1875902 | Zbl 1054.11061
[39] Proximité évanescente. I. La structure polaire d’un -module, Compositio Math., Tome 62 (1987) no. 3, pp. 283-328 | Numdam | MR 901394 | Zbl 0622.32012
[40] Modules d’Alexander et -modules, Duke Math. J., Tome 60 (1990) no. 3, pp. 729-814 | Article | MR 1054533 | Zbl 0715.14007
[41] Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra, Tome 100 (1995) no. 1-3, pp. 93-102 | Article | MR 1344845 | Zbl 0849.32025
[42] Subspaces of moduli spaces of rank one local systems, Ann. Sci. École Norm. Sup. (4), Tome 26 (1993) no. 3, pp. 361-401 | Numdam | MR 1222278 | Zbl 0798.14005