Singularities of $J$ -holomorphic curves

Abstract

Let (V , J ) be an almost complex manifold. Recall that a map from a Riemann surface 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 , v0) is topologically equivalent to a standard holomorphic germ from (C, 0) to (Cn , 0). This was then strengthened to a C 1equivalence 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: