MathDBYau–Tian–Donaldson conjecture for constant scalar curvature metrics
Yau–Tian–Donaldson conjecture for constant scalar curvature metrics
For every polarized smooth projective complex manifold , the polarized manifold is -polystable if and only if the Kähler class admits a constant scalar curvature Kähler metric on .
Progress summary
An unrefereed August 2026 preprint claims to have found a counterexample showing that the conjectured link between algebraic stability and constant-curvature metrics is false.
The conjecture asserts that a polarized manifold admits a constant scalar curvature Kähler metric exactly when it is -polystable. Earlier work established equivalences for strengthened or modified stability notions, but the ordinary correspondence remained open in general.
August 2026 claimed counterexample
Liu Jihao’s preprint claims an explicit counterexample to the ordinary equivalence between -polystability and existence of a cscK metric. If correct, this would refute the conjecture; the claim is currently unrefereed and independently unconfirmed.
Current status (as of August 2026): The ordinary Yau–Tian–Donaldson equivalence remains unsettled, with a new unverified preprint claiming its refutation and no recorded independent confirmation.
A preprint claims to disprove the Yau–Tian–Donaldson conjecture
A preprint claims to disprove the Yau–Tian–Donaldson conjecture
A preprint claims to disprove the Yau–Tian–Donaldson conjecture
Sources
Sources & referencesView supporting material
Primary source
Additional references
- Disproof of the Yau--Tian--Donaldson conjecture — arXiv — Liu, Jihao
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