Algebraic & Geometric Topology

Nongeneric $J$–holomorphic curves and singular inflation

Dusa McDuff and Emmanuel Opshtein

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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; ) 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.

Article information

Algebr. Geom. Topol., Volume 15, Number 1 (2015), 231-286.

Received: 3 December 2013
Revised: 16 June 2014
Accepted: 18 June 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]

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


McDuff, Dusa; Opshtein, Emmanuel. Nongeneric $J$–holomorphic curves and singular inflation. Algebr. Geom. Topol. 15 (2015), no. 1, 231--286. doi:10.2140/agt.2015.15.231.

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