Images of analytic map germs and singular fibrations
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Images of analytic map germs and singular fibrations Cezar Joi¸ta1 · Mihai Tibar ˘ 2 In memoriam S¸ tefan Papadima Received: 3 March 2019 / Revised: 17 July 2019 / Accepted: 4 September 2019 © Springer Nature Switzerland AG 2019
Abstract For a map germ G with target (C p, 0) or (R p, 0) with p 2, we address two phenomena which do not occur when p = 1: the image of G may be not well-defined as a set germ, and a local fibration near the origin may not exist. We show how these two phenomena are related and how they can be characterised. Keywords Map germs · Local fibrations · Images of analytic maps Mathematics Subject Classification 14D06 · 32S55 · 58K05 · 57R45 · 14P10 · 32S20 · 32S60 · 58K15 · 57Q45 · 32C40
1 Introduction We focus here on two phenomena concerning analytic map germs (Km, 0) → (K p, 0) with p 2, where K = R or C, which do not occur when p = 1: (A) The image of a map germ may be not well-defined as a set germ. (B) A map germ may not define a local fibration. In support of the assertion (A), one of the simplest examples is the blow-up F : (K2, 0) → (K2, 0), F(x, y) = (x, x y). The image F(Bε ) of the ball Bε cen-
The authors acknowledge the support of the Labex CEMPI (ANR-11-LABX-0007-01). The first author acknowledges the CNCS Grant PN-III-P4-ID-PCE-2016-0330.
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Cezar Joi¸ta [email protected] Mihai Tib˘ar [email protected]
1
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
2
CNRS, UMR 8524 – Laboratoire Paul Painlevé, Université de Lille, 59000 Lille, France
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C. Joi¸ta, M. Tib˘ar
tred at 0, viewed as a set germ at 0, depends heavily on the radius ε > 0, and therefore the image of the map germ F is not well-defined as a set germ. The germ-image condition was singled out by Mather [28, Sections 2.5 and 9]1 as a necessary condition for the construction of a certain Whitney stratification. However, in the study of map germs initiated by Thom, Milnor, Mather, Arnold, etc. and continued by many mathematicians until today, the usual setting is “map germ G with isolated singularity in its central fibre G −1 (0)”, and in this setting the germ-image condition turns out to be fulfilled, both over C and over R—see below our Propositions 2.3 and 2.4 which treat more general settings. Our new study concerns the complementary case “nonisolated singularities in the central fibre”. In support of the assertion (B), one may consider the following example taken from [39]: F : (C3, 0) → (C2, 0), (x, y, z) → (x 2 − y 2 z, y). Then F is flat, so its image is well-defined as a set germ: (Im F, 0) = (C2, 0); see, e.g. [7, p. 214]. Sabbah [39] showed that this map germ, with Sing F = {x = y = 0} and F(Sing F) = {0}, does not have a locally trivial fibration over the set germ (C2 \ {0}, 0). The following natural questions arise: How can one characterise the phenomena (A) and (B)? Are they related? We address here (A) and (B), in this order, for the following reason: the existence of a well-defined image of a map germ is a necessary condition for
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