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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.

Sources for this record

What was already known

Boucksom, Guedj and Lu asked whether every compact complex manifold has bounded mass, specifically noting Hopf manifolds of dimension at least three. The property was known in complex dimension at most two and for Fujiki-class manifolds; the Hopf threefold was an open test case.

AI involvement AI-led

The paper credits the initial counterexample to a Rethlas agent using GPT-5.6 Sol. Felix Ye improved it; Mingchen Xia and Kewei Zhang simplified the construction and explain the final mass and heat-kernel argument.

The initial counterexample itself, rather than peripheral assistance, is explicitly credited to an AI agent; later human improvements and simplification are separately credited.

Importance Notable

The Hopf threefold example answers a named bounded-mass question in complex geometry.

Limitations

The authors do not document a completed technical check of the final theorem or identify which final lemmas came from the initial agent construction.

Technical verification Reported

The paper supplies a detailed proof and the authors accept responsibility, but the available disclosure does not document a completed human or machine technical check of Theorem 1.1. Reported is the evidence default.

Human understanding Full

The human authors simplified the construction and explain the exact mass identity, positive-potential construction and diverging fiber energy in the final manuscript.

Correction and publication history