Nongeneric J–holomorphic curves and singular inflation

McDuff, Dusa; Opshtein, Emmanuel

doi:10.2140/agt.2015.15.231

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Nongeneric $J$–holomorphic curves and singular inflation

Dusa McDuff and Emmanuel Opshtein

Algebraic & Geometric Topology 15 (2015) 231–286

DOI: 10.2140/agt.2015.15.231

Abstract

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 $A \in H_{2} (M; ℤ)$ 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.

Keywords

$J$–holomorphic curve, rational symplectic $4$–manifold, negative divisor, relative symplectic inflation, relative symplectic cone

Mathematical Subject Classification 2010

Primary: 53D35

References

Publication

Received: 3 December 2013

Revised: 16 June 2014

Accepted: 18 June 2014

Published: 23 March 2015

Authors

Dusa McDuff
Department of Mathematics, Barnard College Columbia University 2990 Broadway New York, NY 10027 United States

Emmanuel Opshtein
IRMA Université de Strasbourg 67000 Strasbourg France

Volume 15, issue 1 (2015)