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Bounded mass property for compact complex manifolds

A compact complex manifold XX of dimension nn has the bounded mass property if, for one (equivalently every) Hermitian form ω\omega, the Monge-Ampère masses ∫X(ω+ddcφ)n\int_X(\omega+dd^c\varphi)^n are uniformly bounded over all smooth φ\varphi with ω+ddcφ>0\omega+dd^c\varphi>0. On a compact Kähler manifold Stokes' theorem makes that mass independent of φ\varphi outright; for a merely Hermitian ω\omega, which is not closed, it genuinely depends on φ\varphi, and controlling it is a recurring theme of Hermitian pluripotential theory. The property is known to hold in dimension n≤2n\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=(C3∖{0})/⟨z↦e−1z⟩X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Construction
Field
Complex geometry
Posed by
Sébastien Boucksom, Vincent Guedj, Chinh H. Lu
Year posed
2025
Years open
1y
Solved
2026-08-21
Model
Rethlas (GPT-5.6 Sol)
Vendor
OpenAI
Collaborators
Mingchen Xia, Kewei Zhang
Verification
Unreviewed
Publication
Preprint
Significance
16 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Disproved on the Hopf threefold X=(C3∖{0})/⟨z↦e−1z⟩X=(\mathbb{C}^3\setminus\{0\})/\langle z\mapsto e^{-1}z\rangle: Xia and Zhang construct a smooth Hermitian form ω\omega and smooth functions φj\varphi_j with ω+ddcφj>0\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.

What the AI did

The paper's acknowledgments, in full: "The initial counterexample was constructed with the Rethlas agent (improved by Felix Ye), using the gpt-5.6-sol model. The authors then simplified the construction and improved the presentation, and are fully responsible for all assertions in this paper." For a disproof the counterexample is the entire result, and the agent produced it; what the authors describe adding is simplification and presentation. So AI-discovered here rests on the authors' own account of what the system did, not on an inference drawn from vague wording. Felix Ye is credited with improving the agent rather than with the mathematics, so he is not listed as a collaborator.

Verification

An arXiv preprint (v1, 21 August 2026, math.CV), unrefereed and with no independent endorsement. No mathematics was checked here, and there is nothing mechanical to check it against - no formalization, no computational certificate. Verified on 25 August 2026: the paper exists at arXiv:2608.21053, and its title, authors, Hopf-threefold statement and date match this entry; its acknowledgments carry the AI disclosure quoted above word for word; and the reference it answers is real - Boucksom, Guedj and Lu, "Volumes of Bott-Chern classes" (arXiv:2406.01090, Peking Math. J. 2025). The paper's introduction is the source for the problem's standing - bounded mass known for n≤2n\le2 and for Fujiki class, the question raised explicitly for Hopf manifolds of dimension at least three in [BGL25, Example 1.19] - which is the authors' characterization of what was open rather than an independent literature search run here.

Sources

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