Positivity of Smooth Currents on Singular Spaces

Abstract

We study two notions of positivity for smooth currents on singular spaces. We show that they are not equivalent by establishing a necessary condition on the fourth Whitney cones. We also construct an explicit compact normal projective variety $X$ and a real smooth $(1,1)$-form $T$ on $X$ such that $T$ has local smooth ddc-potentials and is a Kähler current, but $T$ is not a positive form (hence not Hermitian).