Math. Z. 226, 359–373 (1997) c  Springer-Verlag 1997 Singularities of J -holomorphic curves Jean-Claude Sikorav Universit´ e Paul Sabatier, Laboratoire Emile Picard, URA CNRS 1408, 118 route de Narbonne, F-31062 Toulouse, France (e-mail: sikorav@cict.fr) Received 19 February 1996; in final form 29 February 1996 Introduction Let (V , J ) be an almost complex manifold. Recall that a map from a Riemann sur- face to V is J -holomorphic if its derivative is complex linear. A J -holomorphic curve, or J -curve for short, is the image of such a map [nowhere locally constant]. The study of singularities of J -holomorphic curves began with the statement of M. Gromov [G] about the “positivity of intersections” in the case when V is of dimension 4: the homological intersection of two compact J -curves is a sum of positive local contributions, and a similar property holds for the self-intersection of one curve (adjunction formula). This property is crucial for applications of the theory to symplectic topology in dimension 4 and contact topology in dimension 3 (Gromov, McDuff, Eliashberg, Hofer... ). The positivity of intersections was proved by D. McDuff [McD1] (see also chapter VI in [AL]). Then in [McD2] she proved that in any dimension a germ of J -holomorphic map (C, 0) → (V ,v 0 ) is topologically equivalent to a standard holomorphic germ from (C, 0) to (C n , 0). This was then strengthened to a C 1 - equivalence by M. Micallef and B. White [MW], as a special case of a much more general result on singularities of “generalized minimal” surfaces. They proved also that it applies to reducible germs: Theorem 1 ([MW], Th. 6.2). Let (V , J ) be an almost complex manifold of com- plex dimension n. Let f i :(C, 0) → (V ,v 0 ),i =1,..., r be germs of J - holomorphic maps through the same point. Assume that J is of class C 1,1 . Then there exists a local C 1 -diffeomorphism ϕ :(V ,v 0 ) → (C n , 0), with a complex linear differential d ϕ(v 0 ), and local diffeo- morphisms u i :(C, 0) → (C, 0), tangent to the identity and of class C 2,1− , such that all the ϕ ◦ f i ◦ u i : C → C n are holomorphic. Here we use the notation C k ,1− to mean the intersection of the classes C k ,α for α< 1. It is weaker than C k ,1 which means C k with a Lipschitz k -eth