Geodesic
Périodes évanescentes et (a,b)-modules monogènes
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 651-697.

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In order to describe the asymptotic behaviour of a vanishing period in the degeneration of a one parameter family of complex manifolds, we introduce and use a very simple algebraic structure encoding the corresponding filtered Gauss-Manin connection: regular geometric (a,b)-module generated (as left $\widetilde{A}$-modules) by one element. The idea is to use not the full Brieskorn module associated to the Gauss-Manin connection but the minimal (regular) filtered differential equation satisfied by the period integral we are interested in. We show that the Bernstein polynomial associated is quite simple to compute for such (a,b)-modules and give a precise description of the exponents which appears in the asymptotic expansion which avoids integral shifts. We show the efficiency of this tool on a couple of explicit computations in some classical (but not so easy) examples. 
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Barlet, Daniel. Périodes évanescentes et (a,b)-modules monogènes. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 651-697. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_3_a8/

[A'C.73] N. A'Campo, Sur la monodromie des singularités isolées d'hypersurfaces complexes, Inv. Math., 20 (1973), 147-169. | fulltext EuDML | DOI | MR

[A-G-V] V. Arnold - S. Goussein-Zadé - A. Varchenko, Singularités des applications différentiables, édition MIR, volume 2 (Moscou, 1985).

[Br.70] E. Brieskorn, Die Monodromie der Isolierten Singularita Èten von Hyperflächen, Manuscripta Math., 2 (1970), 103-161. | fulltext EuDML | DOI | MR | Zbl

[B. 93] D. Barlet, Théorie des (a,b)-modules I, in Complex Analysis and Geometry, Plenum Press, (1993), 1-43. | MR | Zbl

[B. 95] D. Barlet, Théorie des (a,b)-modules II. Extensions, in Complex Analysis and Geometry, Pitman Research Notes in Mathematics Series 366 Longman (1997), 19-59. | MR | Zbl

[B. 05] D. Barlet, Module de Brieskorn et forme hermitiennes pour une singularité isolée d'hypersuface, revue de l'Inst. E. Cartan (Nancy), 18 (2005), 19-46. | MR

[B. II] D. Barlet, Sur certaines singularités d'hypersurfaces II, J. Alg. Geom., 17 (2008), 199-254. | DOI | MR | Zbl

[B. 07] D. Barlet, Sur les fonctions a singularité de dimension 1 (version révisée), preprint Institut E. Cartan (Nancy), n. 42 (2008), 1-26, arXiv:0709.0459 (math. CV and math. AG) À paraȋtre au Bulletin de la SMF. | DOI | MR

[B. 08] D. Barlet, Two finiteness theorem for regular (a,b)-modules, preprint Institut E. Cartan (Nancy) n. 5 (2008), 1-38, arXiv:0801.4320 (math. AG and math. CV).

[B.-S. 04] D. Barlet - M. Saito, Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities, J. Lond. Math. Soc. (2), 76 n. 1 (2007), 211-224. | DOI | MR | Zbl

[M. 75] B. Malgrange, Le polynôme de Bernstein d'une singularité isolée, in Lect. Notes in Math., 459 (Springer, 1975), 98-119. | MR

[S. 89] M. Saito, On the structure of Brieskorn lattices, Ann. Inst. Fourier, 39 (1989), 27-72. | fulltext EuDML | MR | Zbl

[Sc.78] J. Scherk, On the Gauss-Manin connectio of an isolated hypersurface singularity, Math. Ann., 238 (1978), 23-32. | fulltext EuDML | DOI | MR | Zbl