The gluing construction for normally generic J-holomorphic curves

TL;DR

This paper introduces a gluing construction for normally generic J-holomorphic curves, showing they form smooth neighborhoods of expected dimension, with applications to the symplectic isotopy problem in dimension 4.

Contribution

It establishes a local Euclidean structure around normally generic J-holomorphic curves and applies this to solve the symplectic isotopy problem for degree 3 surfaces in CP2.

Findings

Normal genericity implies local Euclidean neighborhoods of expected dimension.

In CP2, all curves with only nodes satisfy normal genericity.

Provides a solution to the symplectic isotopy problem for degree 3 surfaces.

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.