A counterexample to the bounded mass property

Abstract

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

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