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K-polystability and existence of cscK metrics

Does K-polystability imply existence of a constant-scalar-curvature Kähler metric on a smooth polarized complex projective variety? The stability notion here tests every normal ample algebraic test configuration of every positive exponent.

Why it matters and what counts as progress

Why it matters: The exact stability definition matters to whether algebraic K-polystability characterizes existence of constant-scalar-curvature Kähler metrics.

What counts as progress: Prove metric existence for a new class or construct a stable polarized variety with no such metric under that exact stability definition.

How the problem developed

  1. 2002 · Conjecture [1][2]

    Simon K. Donaldson Formulated the cscK correspondence with algebraic K-stability for polarized varieties, now called the Yau–Tian–Donaldson conjecture.

  2. 19 Aug 2026 · Counterexample claim [2]

    Jihao Liu Posted a preprint claiming a smooth polarized projective fivefold that is K-polystable for all normal ample algebraic test configurations of every positive exponent but admits no cscK metric.

AI contribution

Sources

  1. Simon K. Donaldson, Scalar curvature and stability of toric varieties
  2. Jihao Liu, Disproof of the Yau–Tian–Donaldson conjecture
  3. Liu’s cscK counterexample paper
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