Nongeneric J-holomorphic curves in symplectic 4-manifolds
Abstract
This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,) when J∈ (), the set of -tame J for which a fixed chain of transversally intersecting embedded spheres of self-intersection -2 is J-holomorphic. Extending work by Biran (in Invent. Math. (1999)), it shows that when (M,) is the blow up of a rational or ruled symplectic 4-manifold, any homology class A∈ H2(M;), with nonzero Gromov invariant and nonnegative intersection both with the spheres in and with the exceptional classes other than A, has an embedded J-holomorphic representative for some J∈ ($. This result is a key step in some of the arguments in McDuff (Journ. Topology (2009)) on embedding ellipsoids, and also has applications to symplectic 4-orbifolds.
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