A Kahlerness criterion for real (1,1)-classes on projective manifolds through extendibility of singular potentials

Let X be a projective manifold and let{θ}∈H 1,1(X,R)be a nonzero pseudo-effective(transcendental)class,where θ is a smooth closed real(1,1)-form.We prove that if for any one-dimensional complex submanifold C⊂X and φ∈SPsh(C,θ|C)with a single analytic singularity at some point p∈C,there exists a function φ∈Psh(X,θ)such that φ|C=φ and φ is continuous at points of C∖{p},then{θ}is a Kahler class.