The local image problem for complex analytic maps

Images of analytic map germs and singular fibrations

“…Proof. This has been stated and proved in [JT2] for X = C N , of course without the interpretation as F * (A X,p ) that we have just introduced above. In the case of a subspace X ⊂ C N , after replacing the balls B ε ⊂ C N centred at the origin by their intersections X ∩ B ε , the proof goes word by word as that in loc.cit.…”

“…one may have the equality of set germs (ImG, 0) = (K 2 , 0), but certain map germs, for instance G(x, y) = (x, xy), do not even have well-defined image as a set germ at the origin. For the problem "when a map germ has a well defined image as a set germ" we refer to [JoT2], see also [ART] and [JoT1].…”

On Singular Maps With Local Fibration

“…The question whether a given map germ has a well-defined image as a set germ seems to be wide open. A first classification has been given in [9] for the case of holomorphic map germs with target (C 2 , 0).…”

Thom property and Milnor–Lê fibration for analytic maps

“…In Section 2, we define the transversality property for an arbitrary $${\mathbb {K}}$$‐analytic map‐germ $$f: (X,0) \rightarrow ({\mathbb {K}}^k,0)$$, which takes into account the first phenomenon mentioned above, regarding the possibility that the image and the discriminant of f may not be well‐defined as set‐germs (Definition 2.1). Then, we give our first result (Theorem 2.2), which says that if f has the transversality property (relative to some stratification $$\mathcal {S}$$ of X ), then its image is well‐defined as a set‐germ, giving an answer to the image problem addressed in [12]. Moreover, f has a Milnor–Lê fibration, and this fibration does not depend on the choices of ε and η.…”

“…Studying the existence of a Milnor–Lê fibration for $${\mathbb {K}}$$‐analytic map‐germs $$f: ({\mathbb {K}}^n,0) \rightarrow ({\mathbb {K}}^k,0)$$ is a very active field of research. We cite [2–8, 11, 12, 16, 17, 19–23, 25] as some examples of the recent development in this area. The same discussion stands for the more general setting of $${\mathbb {K}}$$‐analytic map‐germs $$f: (X,0) \rightarrow ({\mathbb {K}}^k,0)$$, where ( X , 0) is either the germ of a subanalytic set, when $${\mathbb {K}}={\mathbb {R}}$$, or the germ of a complex analytic space, when $${\mathbb {K}}={\mathbb {C}}$$.…”