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\(\mu\)-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series. (English) Zbl 1403.32016

Summary: This paper is a sequel to [C. Hertling, Ann. Inst. Fourier 61, No. 7, 2643–2680 (2011; Zbl 1279.32021)] and [F. Gauss and C. Hertling, in: Singularities and computer algebra. Festschrift for Gert-Martin Greuel on the occasion of his 70th birthday. Based on the conference, Lambrecht (Pfalz), Germany, June 2015. Cham: Springer. 109–146 (2017; Zbl 1380.32029)]. In [Hertling, loc. cit.] a notion of marking of isolated hypersurface singularities was defined, and a moduli space \(M_\mu^{mar}\) for marked singularities in one \(\mu\)-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a \(\mu\)-constant monodromy group \(G^{mar}\subset G_Z\). Here \(G_Z\) is the group of automorphisms of a Milnor lattice which respect the Seifert form.
It was conjectures that \(M^{mar}_\mu\) is connected. This is equivalent to \(G^{mar}=G_{\mathbb{Z}}\). Also Torelli type conjectures were formulated. In [Hertling, loc. cit.] and [Gauss and Hertling, loc. cit.] \(M^{mar}_\mu\), \(G_{\mathbb{Z}}\) and \(G^{mar}\) were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture that \(G^{mar}=G_{Bbb{Z}}\) and \(M^{mar}\) is connected does not hold for certain subseries of the bimodal series.

MSC:

32S15 Equisingularity (topological and analytic)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14D22 Fine and coarse moduli spaces
58K70 Symmetries, equivariance on manifolds
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