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].


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


Zbl 0824.14002