Nongeneric $J$–holomorphic curves and singular inflation

Abstract

This paper investigates the geometry of a symplectic $4$–manifold $\left(M,\omega \right)$ relative to a $J$–holomorphic normal crossing divisor $\mathcal{S}$. Extending work by Biran, we give conditions under which a homology class $A\in H_2\left(M;\mathbb{Z}\right)$ with nontrivial Gromov invariant has an embedded $J$–holomorphic representative for some $\mathcal{S}$–compatible $J$. This holds for example if the class $A$ can be represented by an embedded sphere, or if the components of $\mathcal{S}$ are spheres with self-intersection $-2$. We also show that inflation relative to $\mathcal{S}$ is always possible, a result that allows one to calculate the relative symplectic cone. It also has important applications to various embedding problems, for example of ellipsoids or Lagrangian submanifolds.