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

2011/08/02 by Liat Kessler, Kessler, Liat
Mathematics · #53D20 #53D30 #53D35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1108.0636

openalex publication_date 2011/08/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Symplectic manifolds and their lagrangian submanifolds

Related

• Embedded surfaces for symplectic circle actions
• Generalizing symplectic topology from 1 to 2 dimensions
• Symplectic reduction along a submanifold
• Symplectic embeddings in infinite codimension
• Extension of 2-forms and symplectic varieties