Nongeneric J-holomorphic curves in symplectic 4-manifolds
Abstract
This note discusses the structure of J-holomorphic curves in symplectic 4-manifolds (M,\om) when J\in \Jj(\Ss), the set of \om-tame J for which a fixed chain \Ss of transversally intersecting embedded spheres of self-intersection \le -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\in H_2(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\in \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.
Keywords
Cite
@article{arxiv.1211.2431,
title = {Nongeneric J-holomorphic curves in symplectic 4-manifolds},
author = {Dusa McDuff},
journal= {arXiv preprint arXiv:1211.2431},
year = {2013}
}
Comments
This paper has been withdrawn by the author because its constructions of embedded curves are incorrect, though many other results such as Prop. 1.2.5 and 1.2.7 are correct. The Erratum to the Journ. Topology (2009) papers on embedding ellipsoids explains how to fix the problems with those papers; a new version of this paper, written with E. Opshtein, will appear shortly