Projectivity without a uniformly controlled compatible polarization

Determine whether there exist smooth projective manifolds equipped with Kähler metrics satisfying a uniform Ricci lower bound, diameter bound, and volume noncollapsing, but without a uniformly controlled polarization compatible with the metrics, whose genuinely singular Gromov–Hausdorff limit is not homeomorphic to any complex projective variety.

Background

The paper uses a uniformly controlled polarization to invoke the Liu–Székelyhidi algebraicity theorem, which shows that noncollapsed Gromov–Hausdorff limits of polarized Kähler manifolds with a uniform Ricci lower bound are normal projective varieties. The appendix demonstrates that polarization, noncollapse, and a lower Ricci bound alone do not force uniqueness of metric tangent cones, but its examples still retain a fixed polarization.

The unresolved question is whether the projective-algebraic nature of the limit can fail when the metrics satisfy the stated geometric bounds but no uniformly controlled polarization compatible with them is assumed. The authors also ask whether an analogous phenomenon can occur for conical metrics with a fixed cone angle, but that additional formulation is expressed as an excluded aspiration rather than a separate entry.