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Theory of \((a,b)\)-modules. I. (English) Zbl 0824.14002

Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 1-43 (1993).
The aim of this paper is to develop an elegant algebraic structure which is useful for studying isolated singularities of complex hypersurfaces. Let \(b\) be termwise integration (without constant) on asymptotic expansions at 0 with one variable \(s\), and let the operator \(a\) be multiplication by \(s\). The author constructs the \((a,b)\)-module associated with a simple pole connection, then introduces a notion of regularity. Several examples are given including one showing how to interpret information in case of a \(\mu\)-constant family of isolated singularities in terms of \((a,b)\)-modules.
For the entire collection see [Zbl 0772.00007].

MSC:

14B05 Singularities in algebraic geometry
32S15 Equisingularity (topological and analytic)
14J70 Hypersurfaces and algebraic geometry
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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