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

TL;DR

The paper constructs and analyzes weakly symplectic forms on the space of embedded symplectic surfaces in a symplectic manifold, exploring their reductions and relations to moduli spaces of J-holomorphic curves.

Contribution

It introduces a specific weakly symplectic form on the space of immersions and demonstrates its reduction to a symplectic form on the quotient space, linking it to moduli spaces.

Findings

Construction of a weakly symplectic form on immersion spaces.

Reduction of this form to a symplectic form on the quotient by Hamiltonian actions.

Comparison with other 2-forms and implications for moduli spaces of J-holomorphic curves.

Abstract

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