Rigid and Nonrigid Isotropic Nef Classes on K3 Surfaces

Let $X$ be a fixed complex K3 surface and let $0\neα\in\overline{Kah_X}$ be a $(1,1)$-class on the boundary of the Kähler cone with $α^2=0$. We build a novel connection between two seemingly unrelated, intensively studied problems: the rigidity of a closed positive current in the class $α$ and the c