A counterexample to the bounded mass property

Abstract

A compact complex manifold has the bounded mass property if, for one (equivalently, every) Hermitian form $ω$, the masses $\int_X(ω+\mathrm{dd}^{\mathrm{c}}φ)^n$ are uniformly bounded over all smooth $φ$ with $ω+\mathrm{dd}^{\mathrm{c}}φ>0$. We prove that this property fails on the Hopf threefold $(\mathbb C^3\setminus\{0\})/\langle z\mapsto\mathrm{e}^{-1}z\rangle$, answering a question of Boucksom--Guedj--Lu.