Nongeneric J-holomorphic curves in symplectic 4-manifolds

2012/11/11 by McDuff, Dusa
#53D05 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1211.2431

Abstract

This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J∈ \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss 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,\om) is the blow up of a rational or ruled symplectic 4-manifold, any homology class A∈ H₂(M;\Z), with nonzero Gromov invariant and nonnegative intersection both with the spheres in \Ss and with the exceptional classes other than A, has an embedded J-holomorphic representative for some J∈ \Jj(\Ss. 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.

Related

• Lectures on Gromov invariants for symplectic 4-manifolds
• Invariants of real symplectic 4-manifolds out of reducible and cuspidal curves
• Symplectic surfaces and generic J-holomorphic structures on 4-manifolds
• Holomorphic spheres and four-dimensional symplectic pairs
• Disques J-holomorphes contenus dans une hypersurface