#### Volume 15, issue 1 (2015)

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Nongeneric $J$–holomorphic curves and singular inflation

### Dusa McDuff and Emmanuel Opshtein

Algebraic & Geometric Topology 15 (2015) 231–286
##### Abstract

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

##### Keywords
$J$–holomorphic curve, rational symplectic $4$–manifold, negative divisor, relative symplectic inflation, relative symplectic cone
Primary: 53D35