Paper detail

Symplectic forms on the space of embedded symplectic surfaces and their reductions

Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e}(Σ,σ) with respect to the Sympl(Σ,σ)-action. We also compare {ω^{D}}_{(Σ,σ)} and its reduction ω^D_{\red} to another 2-form on the space of immersed symplectic Σ-surfaces in M. We conclude by a result on the restriction of {ω^{D}}_{(Σ,σ)} to moduli spaces of J-holomorphic curves.