Nongeneric J-holomorphic curves and singular inflation

This paper investigates the geometry of a symplectic 4-manifold $(M,\om)$ relative to a J-holomorphic normal crossing divisor S. Extending work by Biran (in Invent. Math. 1999), we give conditions under which a homology class $A\in H_2(M;\Z)$ 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.