2013/09/30 by Dusa McDuff, Emmanuel Opshtein · 1 citation
Mathematics · #math.SG
paper · pdf · doi:10.2140/agt.2015.15.231
published as Algebr. Geom. Topol. 15 (2015) 231-286 · 45 pages. v2: In this version, unnecessary assumptions on the singular set are removed, and a more detailed discussion of 1-parameter families is provided
arxiv created 2013/12/03 · arxiv updated 2015/05/27
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∈ H2(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.