Cauchy–Riemann equations and J-symplectic forms

Let (Σ,j)(Σ,j) be a Riemann surface. The almost complex manifolds (M,J)(M,J) for which the J  -holomorphic curves ϕ:Σ→Mϕ:Σ→M are of variational type, are characterized. This problem is related to the existence of a vertically non-degenerate closed complex 3-form on Σ×MΣ×M (see Theorem 4.3 below), which determines a family of J  -symplectic structures on (M,J)(M,J) parametrized by Σ.