Compatibility of Donaldson–Sun degenerations with log-pluricanonical sheaves

Establish whether the reflexive log-pluricanonical sheaves associated with the Donaldson–Sun degenerations commute with passage to the associated graded object, equivalently whether the specific Donaldson–Sun degenerations remain in a Kollár hull-theoretic stratum where this compatibility holds, so that the degeneration is genuinely mathbb{Q}-Gorenstein.

Background

The paper constructs a logarithmic Donaldson–Sun two-step degeneration from the local germ of a Gromov–Hausdorff limit to an intermediate affine cone and then to a metric tangent cone, while tracking the boundary divisor through the degeneration. To compare this construction with the Li–Xu and Li–Wang–Xu stable-degeneration framework, it is necessary to know that suitable reflexive log-pluricanonical sheaves behave well under passage to the associated graded object.

Kollár’s hull theory provides locally closed strata on which the relevant reflexive hulls are compatible with base change, but the paper does not establish that the particular Donaldson–Sun degenerations lie in one of these strata. The unresolved compatibility is therefore isolated as an additional assumption in the stable-degeneration section.