Yau–Tian–Donaldson conjecture

The conjecture asks whether every polarized smooth projective variety has a constant-scalar-curvature Kähler metric in its polarization exactly when it is $K$-polystable. The full equivalence is distinct from important special cases and modified stability theories.

Known results

• The Fano case, concerning Kähler–Einstein metrics and $K$-polystability, was settled by Chen–Donaldson–Sun, with earlier work by Tian.
• Strong $K$-stability implies existence of a $\mathrm{cscK}$ metric (2013).
• Existence of a $\mathrm{cscK}$ metric implies uniform stability with respect to arcs (2024).
• Modified non-Archimedean stability yields equivalences for weighted $\mathrm{cscK}$ metrics, not the ordinary conjecture (2025).

August 2026 counterexample and competing claim

Jihao Liu, with Bin Dong and Guoxiong Gao, claims a smooth polarized projective fivefold that is $K$-polystable for every normal ample test configuration but has no extremal, hence no $\mathrm{cscK}$, metric. This would disprove the stated conjecture, but the result is unverified. A separate 2026 preprint claims an equivalence using uniform $K$-stability; its relation to the exact ordinary conjecture also remains unconfirmed.

Current status (as of September 2026): The ordinary $K$-polystability/$\mathrm{cscK}$ equivalence remains unsettled; the claimed fivefold counterexample and competing proof claim are unverified.