By M. Manoel, M. C. Romero Fuster, C. T. C. Wall

ISBN-10: 0521169690

ISBN-13: 9780521169691

The biennial conferences at S?o Carlos have helped create a world neighborhood of specialists and younger researchers engaged on singularity conception, with a unique specialize in functions to a wide selection of subject matters in either natural and utilized arithmetic. The 10th assembly, celebrating the sixtieth birthdays of Terence Gaffney and Maria Aparecida Soares Ruas, used to be a special day attracting the simplest recognized names within the sector. This quantity comprises contributions through the attendees, together with 3 articles written or co-authored by means of Gaffney himself, and survey articles at the lifestyles of Milnor fibrations, worldwide classifications and graphs, pairs of foliations on surfaces, and Gaffney's paintings on equisingularity.

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2 [5]. Let (V , 0) be a germ of an equidimensional complex analytic space in Cm . Let Vα , α = 1, . . , , be the (connected) strata of a Whitney stratification of a small representative V of (V , 0) such that 0 is in the closure of every stratum. 1 there is ε0 such that for any ε, ε0 > ε > 0 and t0 = 0 sufficiently small, the Euler obstruction of (V , 0) is equal to: χ(Vα ∩ Bε ∩ l −1 (t0 )) · EuV (Vα ), EuV (0) = α=1 where χ denotes the Euler-Poincar´e characteristic and EuV (Vα ) is the value of the Euler obstruction of V at any point of Vα , α = 1, .

Verjovsky. Here we show how stratified Morse theory yields to a clear understanding of what the invariant Euf,V (x) is for arbitrary functions with an isolated singularity. These results can also be deduced from Sch¨urmann’s book [43], and also from the work of D. Massey, as for instance [35, 36]. For this we recall the definition of complex stratified Morse singularities (see Goresky-MacPherson [18], p. 52). 1. Let Vα be a Whitney stratification of V and let f : V → C be the restriction to V of a holomorphic function F : Cm → C; assume for simplicity 0 = f (x).

For 0 < p ≤ d, let f : V → Cp be an analytic map satisfying the condition (δ) for a 2(m − p + 1)-cell σf (that depends on f , [23]) in a cellular decomposition (D) and such that 0 is barycenter of σ . Let v (p) = (v (p−1) , vp ) be a p-frame without singularity on ∂V σ such that vp is a radial vector field in the usual sense, then one has: EuV (Vα ) · χ(Vα ∩ B ∩ f −1 (z0 )) + Euf,V (σ ) Eu(v (p) , V , σ ) = α where {Vα } describes the set of strata in V such that 0 ∈ V α . 12. Local Euler obstruction of 1-forms Let us consider the Nash bundle T on V .

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Real and Complex Singularities by M. Manoel, M. C. Romero Fuster, C. T. C. Wall

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