Nongeneric J–holomorphic curves and singular inflation

McDuff, Dusa¹ ; Opshtein, Emmanuel²

¹Department of Mathematics, Barnard College, Columbia University, 2990 Broadway, New York, NY 10027, United States
²IRMA, Université de Strasbourg, 67000 Strasbourg, France

Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 231-286

Résumé

This paper investigates the geometry of a symplectic 4–manifold (M,ω) relative to a J–holomorphic normal crossing divisor S. Extending work by Biran, 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.

Détail

DOI : 10.2140/agt.2015.15.231

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

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