Volume 15, issue 1 (2015)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 4, 1827–2458
Issue 3, 1253–1825
Issue 2, 621–1251
Issue 1, 1–620

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Nongeneric $J$–holomorphic curves and singular inflation

Dusa McDuff and Emmanuel Opshtein

Algebraic & Geometric Topology 15 (2015) 231–286

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.

$J$–holomorphic curve, rational symplectic $4$–manifold, negative divisor, relative symplectic inflation, relative symplectic cone
Mathematical Subject Classification 2010
Primary: 53D35
Received: 3 December 2013
Revised: 16 June 2014
Accepted: 18 June 2014
Published: 23 March 2015
Dusa McDuff
Department of Mathematics, Barnard College
Columbia University
2990 Broadway
New York, NY 10027
United States
Emmanuel Opshtein
Université de Strasbourg
67000 Strasbourg