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Abstract Let v be a holomorphic vector field in a neighborhood of a point m 0 in \( {\Bbb C}^2 \), which is a non dicritical isolated singularity. Let f = 0 be a reduced equation of the maximal separatrix V through m 0, v f the vector field \( v_f = f '_y {\partial \over \partial x} - f '_x {\partial \over \partial y} \), and \( \Sigma \) the union of separatrices and pseudo-separatrices (i.e. the set of points where v and v f are colinear). Assuming the foliations defined by v and v f to be distinct, we prove that the Baum-Bott residue BB(c 1 2 , v) of v at m 0, as well as the difference PH(v) - μ between the Poincaré-Hopf index and the Milnor number of V at m 0, are "localised" near the separatrices and pseudo-separatrices. (The particular case of generalized curves has already been studied in details in [CLS] and [Br]). We also interpret in K-theory the difference PH - μ as well as the GSV index of Gomez Mont-Seade-Verjovski, and we give a caracterisation of generalized curves in this framework, which will enable us to extend this concept in higher dimension.

Localisation des résidus de Baum-Bott, courbes généralisées et K-théorie (I: feuilletages dans $ {\Bbb C}^2 $ )