Abstract

Let $(M,\omega )$ be a symplectic manifold, and $(\mathrm{\Sigma },\sigma )$ a closed connected symplectic $2$-manifold. We construct a weakly symplectic form ${\omega }^D$ on ${C<!--FUNCTION\; APPLICATION-->}^{\infty }(\mathrm{\Sigma },M)$ which is a special case of Donaldson’s form. We show that the restriction of ${\omega }^D$ to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding $(\mathrm{\Sigma },\sigma )\hookrightarrow (M,\omega )$ descends to a weakly symplectic form on the quotient by $\mathrm{Sympl}<!--FUNCTION\; APPLICATION-->(\mathrm{\Sigma },\sigma )$, and that the symplectic space obtained is a symplectic quotient of the subspace of symplectic embeddings with respect to the $\mathrm{Sympl}<!--FUNCTION\; APPLICATION-->(\mathrm{\Sigma },\sigma )$-action. We also compare ${\omega }^D$ to another $2$-form. We conclude with a result on the restriction of ${\omega }^D$ to moduli spaces of holomorphic curves.

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