The gluing construction for normally generic J-holomorphic curves

Abstract

Under an assumption of normal genericity, we show that a stable J-holomorphic curve has, in the space of homologous curves of the same genus, a locally Euclidean neighbourhood of the expected dimension given by Riemann-Roch. In dimension 4, the normal genericity condition is satisfied in by every curve in CP2 (for an almost complex structure homotopic with the standard one) which has only nodes as singularities. This leads in particular to a solution of the symplectic isotopy problem for surfaces of degree 3.