#### Vol. 316, No. 2, 2022

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A symplectic form on the space of embedded symplectic surfaces and its reduction by reparametrizations

### Liat Kessler

Vol. 316 (2022), No. 2, 409–430
##### Abstract

Let $\left(M,\omega \right)$ be a symplectic manifold, and $\left(\mathrm{\Sigma },\sigma \right)$ a closed connected symplectic $2$-manifold. We construct a weakly symplectic form ${\omega }^{D}$ on ${C}^{\infty }\left(\mathrm{\Sigma },M\right)$ 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 $\left(\mathrm{\Sigma },\sigma \right)↪\left(M,\omega \right)$ descends to a weakly symplectic form on the quotient by $\mathrm{Sympl}\left(\mathrm{\Sigma },\sigma \right)$, and that the symplectic space obtained is a symplectic quotient of the subspace of symplectic embeddings with respect to the $\mathrm{Sympl}\left(\mathrm{\Sigma },\sigma \right)$-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.

##### Keywords
symplectic manifold, Donaldson's form, symplectic quotient, moduli space of J-holomorphic curves
##### Mathematical Subject Classification
Primary: 32Q60, 32Q65, 53D30, 53D35, 58B99