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A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors. (English) Zbl 1327.14090
Summary: In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the \(\mathcal{D} [s]\)-module \(\mathcal{D} [s] h^s\) admits a Spencer logarithmic resolution satisfies the symmetry property \(b(- s - 2) = \pm b(s)\). This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection \(\mathcal{E}\) and of its dual \(\mathcal{E}^\ast\) with respect to a free divisor of linear Jacobian type are related by the equality \(b_{\mathcal{E}}(s) = \pm b_{\mathcal{E}^\ast}(- s - 2)\). Our results are based on the behaviour of the modules \(\mathcal{D} [s] h^s\) and \(\mathcal{D} [s] \mathcal{E} [s] h^s\) under duality.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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[1] Andres, D., Noncommutative computer algebra with applications in algebraic analysis, (2013), Aachen University, Ph.D.
[2] Bernstein, J., The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl., 6, 26-40, (1972)
[3] Bruns, W.; Vetter, U., Determinantal rings, Lecture Notes in Math., vol. 1327, (1988), Springer-Verlag Berlin, Heidelberg · Zbl 0673.13006
[4] Calderón-Moreno, F. J., Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. Éc. Norm. Supér. (4), 32, 5, 701-714, (1999) · Zbl 0955.14013
[5] Calderón-Moreno, F. J.; Mond, D. Q.; Narváez-Macarro, L.; Castro-Jiménez, F. J., Logarithmic cohomology of the complement of a plane curve, Comment. Math. Helv., 77, 1, 24-38, (2002) · Zbl 1010.32016
[6] Calderón-Moreno, F. J.; Narváez-Macarro, L., The module \(\mathcal{D} f^s\) for locally quasi-homogeneous free divisors, Compos. Math., 134, 1, 59-74, (2002) · Zbl 1017.32023
[7] Moreno, F. J.C.; Narváez Macarro, L., Dualité et comparaison sur LES complexes de de Rham logarithmiques par rapport aux diviseurs libres, Ann. Inst. Fourier (Grenoble), 55, 1, 47-75, (2005) · Zbl 1089.32003
[8] Moreno, F. J.C.; Narváez Macarro, L., A mixed associativity formula for tensor products over two Lie-rinehart algebras, Ann. Univ. Ferrara Sez. VII Sci. Mat., LI, 105-118, (2005) · Zbl 1136.17008
[9] Moreno, F. J.C.; Narváez Macarro, L., On the logarithmic comparison theorem for integrable logarithmic connections, Proc. Lond. Math. Soc. (3), 98, 585-606, (2009) · Zbl 1166.32005
[10] Castro-Jiménez, F. J.; Mond, D.; Narváez-Macarro, L., Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc., 348, 3037-3049, (1996) · Zbl 0862.32021
[11] Castro-Jiménez, F. J.; Ucha-Enríquez, J. M., Free divisors and duality for \(\mathcal{D}\)-modules, Proc. Steklov Inst. Math., 238, 88-96, (2002) · Zbl 1039.32011
[12] Castro-Jiménez, F. J.; Ucha-Enríquez, J. M., Testing the logarithmic comparison theorem for free divisors, Exp. Math., 13, 4, 441-449, (2004) · Zbl 1071.14024
[13] Chemla, S., A duality property for complex Lie algebroids, Math. Z., 232, 2, 367-388, (1999) · Zbl 0933.32015
[14] Esnault, H.; Viehweg, E., Logarithmic de Rham complexes and vanishing theorems, Invent. Math., 86, 161-194, (1986) · Zbl 0603.32006
[15] Granger, M.; Mond, D.; Nieto-Reyes, A.; Schulze, M., Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble), 59, 2, 811-850, (2009) · Zbl 1163.32014
[16] Granger, M.; Schulze, M., On the symmetry of b-functions of linear free divisors, Publ. RIMS Kyoto Univ., 46, 479-506, (2010) · Zbl 1202.14046
[17] Huebschmann, J., Poisson cohomology and quantization, J. Reine Angew. Math., 408, 57-113, (1990) · Zbl 0699.53037
[18] Huebschmann, J., Duality for Lie-rinehart algebras and the modular class, J. Reine Angew. Math., 510, 103-159, (1999) · Zbl 1034.53083
[19] Huneke, C., Determinantal ideals of linear type, Arch. Math. (Basel), 47, 4, 324-329, (1986) · Zbl 0613.14037
[20] Kashiwara, M., B-functions and holonomic systems, Invent. Math., 38, 33-53, (1976) · Zbl 0354.35082
[21] Kashiwara, M., On the holonomic systems of linear differential equations, II, Invent. Math., 49, 121-135, (1978) · Zbl 0401.32005
[22] Looijenga, E. J.N., Isolated singular points on complete intersections, London Mathem. Soc. Lect. Notes Series, vol. 77, (1984), Cambridge University Press Cambridge · Zbl 0552.14002
[23] Malgrange, B., Le polynôme de Bernstein d’une singularité isolée, (Fourier Integral Operators and Partial Differential Equations, Colloq. Internat., Univ. Nice, Nice, 1974, Lecture Notes in Math., vol. 459, (1975), Springer Berlin), 98-119
[24] Mather, J. N., Stability of \(C^\infty\) mappings. VI: the Nice dimensions, (Proceedings of Liverpool Singularities-Symposium, I, 1969/1970, Lecture Notes in Math., vol. 192, (1971), Springer Berlin), 207-253
[25] Mebkhout, Z.; Narváez-Macarro, L., La théorie du polynôme de Bernstein-Sato pour LES algèbres de Tate et de dwork-Monsky-washnitzer, Ann. Sci. Éc. Norm. Supér. (4), 24, 2, 227-256, (1991) · Zbl 0765.14009
[26] Micali, A., Sur LES algèbres universelles, Ann. Inst. Fourier (Grenoble), 14, 2, 33-88, (1964) · Zbl 0152.02602
[27] Nakayama, H.; Sekiguchi, J., Determination of b-functions of polynomials defining Saito free divisors related with simple curve singularities of types \(E_6, E_7, E_8\), Kumamoto J. Math., 22, 1-15, (2009) · Zbl 1181.14006
[28] Narváez Macarro, L., Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors, (Singularities I, Algebraic and Analytic Aspects, International Conference in Honor of the 60th Birthday of Lê Dũng Tráng, January 8-26, 2007, Cuernavaca, México, Contemp. Math., vol. 474, (2008), AMS), 245-269 · Zbl 1166.32006
[29] L. Narváez Macarro, Integrable logarithmic connections with respect to a cusp, Course at the Programme “Algebraic versus Analytic Geometry”, Erwin Schrödinger International Institute for Mathematical Physics, November 19-December 13, 2011, Vienna, Austria.
[30] Rinehart, G. S., Differential forms on general commutative algebras, Trans. Amer. Math. Soc., 108, 195-222, (1963) · Zbl 0113.26204
[31] Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, 27, 265-291, (1980) · Zbl 0496.32007
[32] Sevenheck, C., Bernstein polynomials and spectral numbers for linear free divisors, Ann. Inst. Fourier (Grenoble), 61, 1, 379-400, (2011) · Zbl 1221.34237
[33] Teissier, B., Cycles évanescents, sections planes et conditions de Whitney, (Singularités à Cargèse, Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972, Astérisque, vol. 7-8, (1973), Soc. Math. France Paris), 285-362 · Zbl 0295.14003
[34] Torrelli, T., Polynômes de Bernstein associés à une fonction sur une intersection complète à singularité isolée, Ann. Inst. Fourier (Grenoble), 52, 1, 221-244, (2002) · Zbl 1015.32009
[35] Torrelli, T., On meromorphic functions defined by a differential system of order 1, Bull. Soc. Math. France, 132, 591-612, (2004) · Zbl 1080.32011
[36] Ucha Enríquez, J. M., Métodos constructivos en álgebras de operadores diferenciales, (September 1999), Univ. Sevilla, Ph.D.
[37] Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2, (1998), Springer Verlag New York
[38] Yano, T., On the theory of b-functions, Publ. RIMS Kyoto Univ., 14, 111-202, (1978) · Zbl 0389.32005
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