Reflection maps
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Mathematische Annalen
Reflection maps Guillermo Peñafort Sanchis1 Received: 18 June 2019 / Revised: 1 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a reflection group G acting on a complex vector space V , a reflection map is the composition of an embedding X → V with the quotient map V → C p of G. We show how these maps, which can highly singular, may be studied in terms of the group action. We give obstructions to A-stability and A-finiteness of reflection maps and produce, in the unobstructed cases, infinite families of finitely determined map-germs of any corank. We relate these maps to conjectures of Lê, Mond and Ruas.
1 Introduction The aim of this work is to show how to use complex reflection groups to produce singular map-germs (Cn , 0) → (C p , 0), with p > n, and to relate the geometry of the resulting maps to the group action. As we will see, this gives us tools to study a class of map-germs—the reflection maps of the title—containing very degenerate objects, in the sense that they have high corank and multiplicity. Before explaining what reflection maps are, let us give a quick overview of the notions about singularities of mappings most related to this work. A general reference on the topic is [38]. In the field of singularities of mappings, it is customary to identify mappings that go one into the other by means of changes of coordinates. More precisely, two map-germs f , f : (Cn , S) → (C p , 0) are A-equivalent if there exist germs of bi-holomorphism ψ: (Cn , S) → (Cn , S) and φ: (C p , 0) → (C p , 0), such that f = φ ◦ f ◦ ψ.
Communicated by Jean-Yves Welschinger. Work partially supported by the Basque Government through the BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718, by the ERCEA Consolidator Grant 615655 NMST, by CNPq project CSF 401947/2013-0 and FAPESP project 2016/23906-3.
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Guillermo Peñafort Sanchis [email protected] Basque Center for Applied Mathematics, Bilbao, Spain
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G. Peñafort Sanchis
The simplest map-germs are the stable map-germs, whose A-class does not change after a small perturbation. The precise definition is a bit more technical: An unfolding of a multi-germ f : (Cn , S) → (C p , 0) is a map-germ of the form F = (t, f t (x)): (Cr +n , {0} × S) → (Cr + p , 0), with f 0 = f . Un unfolding F is called A-trivial if it is A-equivalent to idCr × f as an unfolding (that is, A-equivalent via unfoldings of the identity maps on (Cn , S) and (C p , 0)). Definition 1 A multi-germ f : (Cn , S) → (C p , 0) is A-stable if every unfolding of f is A-trivial. A finite map f : X → Y is locally A-stable if, for all y ∈ Y , the multi-germ of f at f −1 (y) is A-stable. Since no equivalence relation other than A-equivalence is used and we do not study global stability, here local A-stability is just called stability. Next in complexity, after the stable A-classes, are the map-germs with isolated instability. As i
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