Algebraic & Geometric Topology

Nongeneric $J$–holomorphic curves and singular inflation

Dusa McDuff and Emmanuel Opshtein

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510840912

Digital Object Identifier
doi:10.2140/agt.2015.15.231

Mathematical Reviews number (MathSciNet)
MR3325737

Zentralblatt MATH identifier
1328.53106

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

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

Citation

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. https://projecteuclid.org/euclid.agt/1510840912


Export citation

References

  • P Biran, A stability property of symplectic packing, Invent. Math. 136 (1999) 123–155
  • M,S Borman, T-J Li, W Wu, Spherical Lagrangians via ball packings and symplectic cutting, Selecta Math. 20 (2014) 261–283
  • O Buşe, Negative inflation and stability in symplectomorphism groups of ruled surfaces, J. Symplectic Geom. 9 (2011) 147–160
  • O Buse, R Hind, Symplectic embeddings of ellipsoids in dimension greater than four, Geom. Topol. 15 (2011) 2091–2110
  • S,K Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996) 666–705
  • J,G Dorfmeister, T-J Li, The relative symplectic cone and $T\sp 2$–fibrations, J. Symplectic Geom. 8 (2010) 1–35
  • H Hofer, V Lizan, J-C Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997) 149–159
  • P,B Kronheimer, T,S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797–808
  • F Lalonde, Isotopy of symplectic balls, Gromov's radius and the structure of ruled symplectic $4$–manifolds, Math. Ann. 300 (1994) 273–296
  • B-H Li, T-J Li, Symplectic genus, minimal genus and diffeomorphisms, Asian J. Math. 6 (2002) 123–144
  • T-J Li, A-K Liu, General wall crossing formula, Math. Res. Lett. 2 (1995) 797–810
  • T-J Li, A-K Liu, The equivalence between ${\rm SW}$ and ${\rm Gr}$ in the case where $b\sp +=1$, Internat. Math. Res. Notices (1999) 335–345
  • T-J Li, A-K Liu, Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with $B\sp +=1$, J. Differential Geom. 58 (2001) 331–370
  • T-J Li, M Usher, Symplectic forms and surfaces of negative square, J. Symplectic Geom. 4 (2006) 71–91
  • T-J Li, W Zhang, J–holomorphic curves in a nef class
  • D McDuff, The local behaviour of holomorphic curves in almost complex $4$–manifolds, J. Differential Geom. 34 (1991) 143–164
  • D McDuff, Singularities and positivity of intersections of J–holomorphic curves, from: “Holomorphic curves in symplectic geometry”, Progr. Math. 117, Birkhäuser, Basel (1994) 191–215
  • D McDuff, Lectures on Gromov invariants for symplectic $4$–manifolds, from: “Gauge theory and symplectic geometry”, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, Kluwer, Dordrecht (1997) 175–210
  • D McDuff, From symplectic deformation to isotopy, from: “Topics in symplectic $4$–manifolds”, Int. Press, Cambridge, MA (1998) 85–99
  • D McDuff, Symplectic embeddings of $4$–dimensional ellipsoids, J. Topol. 2 (2009) 1–22
  • D McDuff, The Hofer conjecture on embedding symplectic ellipsoids, J. Differential Geom. 88 (2011) 519–532
  • D McDuff, Symplectic embeddings of $4$–dimensional ellipsoids: Erratum, preprint (2013)
  • D McDuff, L Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994) 405–434
  • D McDuff, D Salamon, A survey of symplectic $4$–manifolds with $b\sp {+}=1$, Turkish J. Math. 20 (1996) 47–60
  • D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, 2nd edition, AMS Colloq. Pub. 52, Amer. Math. Soc. (2012)
  • D McDuff, F Schlenk, The embedding capacity of $4$–dimensional symplectic ellipsoids, Ann. of Math. 175 (2012) 1191–1282
  • E Opshtein, Singular polarizations and ellipsoid packings, Int. Math. Res. Not. 2013 (2013) 2568–2600
  • C,H Taubes, The Seiberg–Witten and Gromov invariants, Math. Res. Lett. 2 (1995) 221–238
  • C,H Taubes, Counting pseudo-holomorphic submanifolds in dimension $4$, J. Differential Geom. 44 (1996) 818–893
  • W Weiwei, Exact Lagrangians in $A_n$ surface singularities