## The theory of $$(a,b)$$-modules. II: Extensions. (Théorie des $$(a,b)$$-modules. II: Extensions.)(French)Zbl 0935.32023

Ancona, V. (ed.) et al., Complex analysis and geometry. Proceedings of the CIRM conferences on complex analysis and geometry XII, Trento, Italy, June 5-9, 1995 and vector bundles on Fano threefolds II, Trento, Italy, December 2-5, 1995. Bonn: Longman. Pitman Res. Notes Math. Ser. 366, 19-59 (1997).
From the introduction (translated from the French): “Based on the notion of $$(a,b)$$-module introduced in part I of this paper [D. Barlet, Complex analysis and geometry, 1-43 (1993; Zbl 0824.14002)], the author studies the filtred Gauss-Manin system associated to a singularity of a holomorphic function. He considers here the $$(a,b)$$-modules as modules over the $$\mathbb{C}$$-algebra $$A:=\oplus_{n\geq 0}\mathbb{C} [[b]]_a^n$$ (where $$ab=ba=a^2$$) with $$b$$-adic completion $$\widetilde A$$. He shows that the Ext of two $$(a,b)$$-modules are vector spaces of finite dimension, a result which is generalized to invariables in $$\S 6$$.
In $$\S 5$$ the author gives useful constructions for defininig the notion of mixed Hodge structure on a geometric $$(a,b)$$-module similar to the construction of A. Varchenko in the case of an isolated singularity”.
For the entire collection see [Zbl 0869.00034].

### MSC:

 32S25 Complex surface and hypersurface singularities 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects)

Zbl 0824.14002