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 (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form ωD on C (Σ,M) which is a special case of Donaldson’s form. We show that the restriction of ωD to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ)(M,ω) descends to a weakly symplectic form on the quotient by Sympl (Σ,σ), and that the symplectic space obtained is a symplectic quotient of the subspace of symplectic embeddings with respect to the Sympl (Σ,σ)-action. We also compare ωD to another 2-form. We conclude with a result on the restriction of ωD 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
Milestones
Received: 1 July 2019
Revised: 27 May 2021
Accepted: 27 November 2021
Published: 6 April 2022
Authors
Liat Kessler
Department of Mathematics, Physics, and Computer Science
University of Haifa
Haifa
Israel