Bounded mass property for compact complex manifolds

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.

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.