Bounded mass property on compact complex manifolds

Does every compact complex manifold have bounded mass for positive forms in a fixed Hermitian potential class? The focused test is the Hopf threefold X=(C³∖{0})/⟨z↦e^−1z⟩: can smooth ω and φⱼ satisfy ω+ddᶜφⱼ>0 while ∫X(ω+ddᶜφⱼ)³ is unbounded?

Why it matters and what counts as progress

Why it matters: It asks whether a finiteness property known in complex dimension at most two and for Fujiki-class manifolds extends to a basic non-Kähler threefold.

What counts as progress: Construct uniformly positive perturbed Hermitian forms with unbounded total top-degree mass, or identify classes where a uniform mass bound survives.

How the problem developed

2024 · Open question [1]
Boucksom, Guedj, and Lu Introduced the bounded mass property in their study of volumes of Bott–Chern classes and asked whether it holds on Hopf manifolds of dimension at least three.
21 Aug 2026 · Counterexample [2]
Mingchen Xia and Kewei Zhang Showed that the bounded mass property fails on the Hopf threefold by constructing positive forms with unbounded mass.

AI contribution

A counterexample to the bounded mass property

Mingchen Xia, Kewei Zhang

For the Hopf threefold X=(C³\{0})/⟨z↦e^-1 z⟩, Xia and Zhang’s Theorem 1.1 constructs a smooth Hermitian form ω and smooth potentials φ_j with ω+dd^cφ_j>0 yet ∫_X(ω+dd^cφ_j)^3→∞. Thus this compact complex manifold lacks the bounded mass property. The paper’s proof uses an elliptic fibration, an exact mass identity and fiberwise heat-kernel potentials.

AI involvementAI-led

ImportanceNotable

Technical verificationReported

Human understandingFull