Breakthrough Tracker record

A preprint claims a fivefold counterexample to the cscK Yau–Tian–Donaldson conjecture

Jihao Liu constructs a smooth polarized projective fivefold claimed to be K-polystable but to admit no constant-scalar-curvature Kähler metric. If correct, it disproves the broad cscK form of the Yau–Tian–Donaldson conjecture in dimension five.

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Stable ID: math-yau-tian-donaldson-csck-counterexample-2026
Revision: math-yau-tian-donaldson-csck-counterexample-2026.v1
Field: Mathematics · Algebraic and differential geometry
Evidence: Tier 1 · Peer reviewed: No
Record state: Provisional · Provisional claimed counterexample
Last checked: 2026-08-22

AI role

The manuscript discloses heavy use of GPT-5.6 Sol, Fable 5 and Danus for calculations, proof development and exposition. The author states that he independently verified the output and takes full responsibility for the mathematics.

Record details

Problem or result: The cscK Yau–Tian–Donaldson conjecture for smooth polarized projective varieties
Authors: Jihao Liu
Institutions: Beijing International Center for Mathematical Research, Peking University
Result date: Preprint submitted August 19; manuscript dated August 21, 2026

Why it matters

The conjecture is a central bridge between algebro-geometric stability and canonical Kähler metrics. A valid counterexample would show that K-polystability alone is insufficient for the general cscK problem.

Limits

This is a 79-page first-version preprint with no located peer review, formal verification or independent correctness assessment. Its scope is the broad cscK formulation; it does not undo established Yau–Tian–Donaldson theorems in more restricted Fano or Kähler–Einstein settings.

Sources

Primary: Jihao Liu, Disproof of the Yau–Tian–Donaldson conjecture

Correction and revision history

2026-08-22 — Added after primary-source, scope, status, AI-role and limitation review.

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