An Irrational Crease That Breaks a 40-Year Conjecture
This presentation explains a claimed counterexample to the Yau–Tian–Donaldson conjecture linking K-polystability and constant scalar curvature Kähler metrics. The construction produces a smooth projective fivefold that is K-polystable with respect to every test configuration yet admits no extremal metric, with the obstruction encoded in an irrational double zero of a boundary polynomial. The technical core is a classification theorem proving that every zero-invariant test configuration must be a product, achieved through filtration analysis, convex transforms, and rationality constraints on algebraic measures.Script
For forty years, mathematicians believed that a geometric stability condition called K-polystability should guarantee the existence of metrics with constant scalar curvature. This paper claims to shatter that belief with a single explicit example: a smooth five-dimensional variety that passes every stability test yet cannot support the metric it ought to have.
The construction starts with four curves of wildly different genera: 3 million, 10 thousand, 76, and 46. These numbers are chosen so their Jacobians are pairwise Hom-orthogonal, meaning no nontrivial maps exist between different factors. This orthogonality becomes the key to controlling all possible degenerations later.
The authors construct a polynomial that vanishes at both rational endpoints and has a unique interior zero at the irrational number three minus square root of five over two. This double zero is not a defect; it is the weapon. It allows the construction of approximate constant-scalar-curvature metrics whose quality improves without limit, yet the limiting profile degenerates precisely at this irrational point, blocking any true extremal metric.
The hardest step is proving that every zero-invariant test configuration must be a product. The authors analyze the filtration on section spaces using convex transforms over a Newton body. Rational slices are controlled using Chow stability and multiplication maps, while convexity extends this control to irrational slices. The irrational crease is then excluded because algebraic filtrations have rational spectral data, and a crease at three minus root five over two would produce an irrational coefficient in the Duistermaat measure.
The example is K-polystable but not uniformly so. The authors construct rational test configurations whose crease parameters are Fibonacci ratios converging to the irrational double zero. Each has strictly positive Donaldson–Futaki invariant, but these invariants decay cubically, driving the uniform stability margin to zero. The irrational obstruction is invisible to any individual algebraic test yet approached arbitrarily closely by rational ones.
The conjecture may still hold for varieties with finite automorphism groups. This example has an infinite connected automorphism group generated by fiber scaling, and that symmetry is essential to the nonexistence proof. Whether a counterexample exists without this crutch remains open. If you'd like to explore more cutting-edge results like this one or generate your own research videos, visit EmergentMind.com.
