For every polarized smooth projective complex manifold (X,L), the polarized manifold (X,L) is K-polystable if and only if the Kähler class c_1(L) admits a constant scalar curvature Kähler metric on X.

Yau–Tian–Donaldson conjecture for constant scalar curvature metrics

The conjecture predicts that algebraic stability of a polarized manifold exactly matches the existence of a constant-scalar-curvature Kähler metric. Before the new claim, the general ordinary $K$ -polystability formulation was open, although important special cases and stronger stability versions were known.

Known results

The Fano case was proved by Chen–Donaldson–Sun and Tian.
Chen–Cheng, Darvas–Rubinstein, and Berman–Darvas–Lu related cscK existence to coercivity of Mabuchi’s $K$ -energy.
Stronger uniform or filtration-based stability criteria imply cscK existence.
A May 2026 preprint claims the corresponding equivalence for uniform $K$ -stability, not ordinary $K$ -polystability.

August 2026 claimed counterexample

Jihao Liu’s preprint claims a smooth polarized projective fivefold $(X,A)$ that is ordinary $K$ -polystable but whose class $c_1(A)$ contains no extremal, hence no cscK, metric. If correct, this disproves the conjecture; the manuscript says GPT-5.6-sol, Fable 5, and the Danus system contributed to obtaining the result.

Current status (as of August 2026): The ordinary $K$ -polystability–cscK equivalence is unsettled, with an unverified preprint claiming a counterexample.

Yau–Tian–Donaldson conjecture for constant scalar curvature metrics

Primary source

Additional references

Progress summary

Solutions 0