Wermer Examples and Currents

We give the first examples of positive closed currents T in ${{\mathbb C}^2}We give the first examples of positive closed currents T in \mathbb C²{{\mathbb C}^2} with continuous potentials, T ùT = 0{T \wedge T = 0}, and whose supports do not contain any holomorphic disk. This gives in particular an affirmative answer to a question of Forn?ss and Levenberg. We actually construct examples with potential of class C ^1,α for all α < 1. This regularity is expected to be essentially optimal.