Counterexample to the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics
A smooth polarized fivefold that is K-polystable but admits no constant scalar curvature Kähler metric, disproving the form of the Yau–Tian–Donaldson conjecture that Donaldson stated in 2002.
- Lab
- Independent
- Model
- Claude Fable 5, GPT-5.6-sol and Danus
- Field
- Mathematics
- Date
- 2026-08-19
- Human collaborators
- Jihao Liu, Bin Dong, Guoxiong Gao
- Problem posed
- 2002 · open 24 yrs
What was found
Donaldson conjectured in 2002 that a smooth polarized variety carries a constant scalar curvature Kähler metric exactly when it is K-polystable, the general-polarization successor to the Kähler-Einstein expectation going back to Yau and to Tian's K-stability. This 79-page paper constructs a polarized smooth projective fivefold, proves it K-polystable against all normal ample algebraic test configurations of every positive exponent, and shows it carries no such metric. Its appendix draws a distinction worth keeping: some counterexamples reduce to a finite certificate checkable once the object is written down, as the Jacobian conjecture counterexample does, while others are themselves theorems quantified over everything, and this is the second kind. Candidate manifolds of this shape have been available since 2008; the mathematical content is the classification showing every test configuration has nonnegative Donaldson-Futaki invariant with equality only for products.
Novelty check
The conjecture is named, celebrated and traceable: the paper states it as Donaldson's 2002 conjecture, records the normality repair of the original scheme-theoretic formulation due to Ross and Thomas, and cites the Matsushima and Futaki obstructions and the Chen-Cheng and Berman-Darvas-Sjöström Dinew lines of work as the surrounding literature. No prior counterexample in any dimension appears. The paper is explicit about what was and was not already available: candidate manifolds of the present shape date to 2008, and what was missing is the proof that the mechanism works. The result is new.
Caveats and known objections
A preprint days old at entry, 79 pages, unrefereed and not formalized, and a counterexample of the kind that cannot be checked by evaluating an object, so verification means reading the classification argument. The theorem actually proved is the extremal-metric statement, which is stronger than the cscK wording of the abstract. Autonomy graded collaborative rather than ai-led: the appendix says the author found initial signs of a breakthrough while exploring open problems with Claude Code and then had Claude Code, Codex and Danus work on it in collaboration, with the three systems together producing the counterexample and its proof, but it also records that human input was crucial at one point, when the author recognized the example had to refute either the Codogni-Stoppa conjecture or cscK Yau–Tian–Donaldson and directed the agents to settle which. The appendix further reports that an improved version of Danus, given only the original problem and none of the earlier findings, later settled it alone in 5 hours and 29 minutes without that human input, which would be autonomous; that system is not public and the report is by the same author, so it is not gradeable evidence. The weaker defensible reading applies.
The open problem
MathDBYau–Tian–Donaldson conjecture for constant scalar curvature metrics
383718
Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.
MathDB carries three records for this conjecture under permuted name orders (321390, 331106, 383718). This is the one whose title matches the usual attribution order.
Also recorded at
vibemathedyau-tian-donaldson-conjecture-csck
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Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and Collaborative.
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Graded author verified for verification and collaborative for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-19-yau-tian-donaldsonBibTeX
@misc{whataifound-independent-2026-donaldson,
title = {Counterexample to the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: Collaborative.},
url = {https://whataifound.org/finding/2026-08-19-yau-tian-donaldson}
}