Annals of Global Analysis and Geometry

, Volume 41, Issue 3, pp 265–280 | Cite as

Symplectic geometry on moduli spaces of J-holomorphic curves

Article

Abstract

Let (M, ω) be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) on the space \({\mathcal{S}_{i}(\Sigma)}\) of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M, ω) that ensure that the restriction of \({\omega_{\mathcal{S}_{i}(\Sigma)}}\) to the moduli space of simple immersed J-holomorphic Σ-curves in a homology class \({A \in {H}_2(M,\,\mathbb{Z})}\) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Σ-curves in a given homology class for a generic J.

Keywords

Moduli space Symplectic form J-holomorphic curve Almost Complex structure 

Mathematics Subject Classification (2000)

MSC 53D30 MSC 53D35 MSC 32Q60 MSC 32Q65 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.D. E. Shaw and Co., L.P.New YorkUSA
  2. 2.Department of Mathematics, TechnionHaifaIsrael
  3. 3.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  4. 4.Department of MathematicsWashington UniversitySt. LouisUSA

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