, Volume 2, Issue 3, pp 249-266

Singularities ofJ-holomorphic curves in almost complex 4-manifolds

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This note concerns the structure of singularities of mapsf from a neighborhood of {0} in the complex plane ℂ to an almost complex manifold (V, J), which areJ-holomorphic in the sense thatdf oi =J odf and are singular (i.e.,df = 0) at {0}. The main result is that whenV has dimension 4, the topology of these singularities is the same as in the case whenJ is integrable. Thus, if the image Imf =C is not multiply-covered, there is a neighborhoodU of the pointx = f(0), such that the pair (U, UC) is homeomorphic to the cone overS 3,K x whereK x is an algebraic knot in S3 that depends only on the germC atx.

This paper was partially supported by NSF Grant DMS 9103033. I wish to thank G. Tian and M. Micallef for useful comments.