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.
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Coffey, J., Kessler, L. & Pelayo, Á. Symplectic geometry on moduli spaces of J-holomorphic curves. Ann Glob Anal Geom 41, 265–280 (2012). https://doi.org/10.1007/s10455-011-9281-1
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DOI: https://doi.org/10.1007/s10455-011-9281-1