Nongeneric J-holomorphic curves and singular inflation

This paper investigates the geometry of a symplectic 4-manifold (M, omega) relative to a J-holomorphic normal crossing divisor S. Extending work by Biran, we give conditions under which a homology class A epsilon H-2(M; Z) with nontrivial Gromov invariant has an embedded J-holomorphic representative for some S-compatible J. This holds for example if the class A can be represented by an embedded sphere, or if the components of S are spheres with self-intersection -2. We also show that inflation relative to 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.