Hölder estimates in big and semipositive cohomology classes

Establish whether the uniform Hölder estimates for normalized Kähler potentials and the induced distance functions remain valid when the Kähler class [θ] is big and semipositive rather than Kähler.

Background

For a compact Kähler manifold equipped with a Kähler class [θ], the paper recalls uniform Hölder estimates for normalized Kähler potentials whose Monge–Ampère densities satisfy a fixed L^p bound, together with a corresponding Hölder estimate for the distance functions induced by the associated Kähler metrics. The authors identify extending both estimates to big and semipositive classes as an unresolved issue.

Such classes arise naturally in the study of geometric and analytic structures near singularities, but the loss of strict positivity obstructs the Kähler-geometric methods used for the estimates. The paper notes that subsequent work has made progress on related geometric estimates, without resolving the general question as stated here.