Images of analytic map germs and singular fibrations

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.

Milnor–Hamm Fibration for Mixed Maps

“…More recently, in Joiţa and Tibȃr (2019) the authors showed that when G is a nice map germ then the classical definition and the new Definition 3 agrees.…”

“…In Araújo dos Santos et al (2019a) and Joiţa and Tibȃr (2018) the authors found sufficient conditions to an analytic map germ to be a nice germ and have introduced a good class of maps with this property, namely, the map germ of type fḡ : (C n , 0) → (C, 0) where f , g : (C n , 0) → (C, 0) are holomorphic germs and such that the meromorphic function f /g is irreducible. In Joiţa and Tibȃr (2019), the authors gave a criterion for an analytic map germ to be a nice germ, see Joiţa and Tibȃr (2019, Theorem 4.5).…”

The vanishing cohomology of non‐isolated hypersurface singularities

“…While this result is known (for example, see [49,Corollary 2.3]), for completeness we include here a proof in the spirit of Section 4.3. 𝑏 𝑛 (𝐹) ⩽ 𝜆 0 + 𝑏 𝑛 (ℂlk(𝑋, 0)), (4.33) where 𝜆 0 ∶= 𝜏 𝑓 − 𝜏 𝑙 = int 0 (Γ, 𝑓 −1 (0)) − int 0 (Γ, 𝑙 −1 (0)). † We may refer to [14] for a study of images of map germs in relation with singular fibrations.…”

On the Milnor fibration for $f(\mathbf z)\bar g(\mathbf z)$

“…Pichon and Seade have studied such functions, especially for the case n = 2 ( [25, 26, 27]). There are also works by Fernandez de Bobadilla and Menegon Neto [9], Parameswaran and Tibar [23], Araujo dos Santos, Ribeiro and Tibar [5], Araujo dos Santos, Ribeiro and Tibar [6], and Joita and Tibar [12]. Note that the link of H is the union of two smooth links defined by f and g respectively which intersect transversely along real codimension 2 smooth variety.…”