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

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

2011-08-02

arxiv.org/abs/1108.0636v1

Mathematics

Symplectic Geometry

20 pages

Scientific paper

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.

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