A counterexample to the zero-mass conjecture

Abstract

We construct an explicit negative plurisubharmonic function on the unit polydisk in ${\mathbb{C}}^n,n\ge 2$ with an isolated singularity at the origin. Its Monge-Ampère measure is the Dirac mass at the origin, but its Lelong number vanishes. This gives a negative answer to the zero-mass conjecture of Guedj-Rashkovskii in every complex dimension at least two.