Bounded mass property for compact complex manifolds

A compact complex manifold $X$ of dimension $n$ has the bounded mass property if, for one (equivalently every) Hermitian form $\omega$, the Monge-Ampère masses $\int_X(\omega+dd^c\varphi)^n$ are uniformly bounded over all smooth $\varphi$ with $\omega+dd^c\varphi>0$. On a compact Kähler manifold Stokes' theorem makes that mass independent of $\varphi$ outright; for a merely Hermitian $\omega$, which is not closed, it genuinely depends on $\varphi$, and controlling it is a recurring theme of Hermitian pluripotential theory. The property is known to hold in dimension $n\le2$ and on manifolds of Fujiki class. Boucksom, Guedj and Lu left open whether it holds on every compact complex manifold, raising the question explicitly for Hopf manifolds of dimension at least three. This paper answers it in the negative on the Hopf threefold $X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle$.

Disproved on the Hopf threefold $X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle$: Xia and Zhang construct a smooth Hermitian form $\omega$ and smooth functions $\varphi_j$ with $\omega+dd^c\varphi_j>0$ whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass identity reducing excess Monge-Ampère mass to a fibrewise Dirichlet energy, and heat-kernel regularizations of Green functions that make that energy diverge while preserving positivity. What the negative answer removes is load-bearing rather than incidental: finiteness of this mass is the starting point for the theory of volumes of Bott-Chern classes, and it enters as a standing hypothesis in recent Hermitian pluripotential theory.