Parabolic nef currents on hyperkähler manifolds

LetM be a compact, holomorphically symplectic Kähler manifold, and η a (1,1)-current which is nef (a limit of Kähler forms). Assume that the cohomology class of η is parabolic, that is, its top power vanishes. We prove that all Lelong sets of η are coisotropic. When M is generic, this is used to show that all Lelong numbers of η vanish. We prove that any hyperkähler manifold with Pic(M) = Z has non-trivial coisotropic subvarieties, if a generator of Pic(M) is parabolic.