Algebraic detection of instability for the ACG+08 fourfolds

Determine whether a polarized smooth projective fourfold constructed by Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman in the cited Example 1 admits an algebraic test configuration detecting its instability, despite all of its Ross–Thomas slope degenerations having positive modified Futaki invariant.

Background

The paper discusses earlier candidate counterexamples to the constant-scalar-curvature Yau–Tian–Donaldson conjecture constructed in [ACG+08]. Those fourfolds have an admissible extremal polynomial that is positive at every rational point but has an irrational repeated interior zero, yielding an analytic destabilizing degeneration that is not algebraic.

The unresolved issue is whether some other normal algebraic test configuration detects the instability. The present paper addresses an analogous problem for a newly constructed fivefold, but does not resolve the question for the specific fourfolds from [ACG+08].