Symplectic geometry on moduli spaces of J-holomorphic curves

摘要

Let (M, ω) be a symplectic manifold, and Σ a compact Riemann surface. We define a 2-form ωS _i(σ) on the space S _i(σ) of immersed symplectic surfaces in M, and show that the form is closed and non-degenerate, up to reparametrizations. Then we give conditions on a compatible almost complex structure J on (M, ω) that ensure that the restriction of ωS _i(σ) to the moduli space of simple immersed J-holomorphic Σ-curves in a homology class A ε H _2(M, ?) is a symplectic form, and show applications and examples. In particular, we deduce sufficient conditions for the existence of J-holomorphic Σ-curves in a given homology class for a generic J.