Second countability of the Hermitian–Yang–Mills moduli space

Prove directly that the Hermitian–Yang–Mills moduli space is second countable, thereby determining whether Donaldson’s metrizability theorem can be avoided in the boundedness argument.

Background

The boundedness proof uses Donaldson’s theorem that the relevant smoothable Hermitian–Yang–Mills moduli space is metrizable and compact. The paper notes that metrizability is used to obtain compactness and to control degenerations in the induction argument.

The authors explicitly question whether this dependence can be removed by proving second countability directly. No such proof is supplied, so the issue remains unresolved within the paper.

References

We wonder if it is possible to avoid Donaldson's metrizablity theorem in the argument above. For instance, can we prove directly that the moduli space is second countable?

— Algebraic limits of Hermitian-Yang-Mills connections on Kähler surfaces  (2610.01221 - Zhou, 1 Oct 2026) in Remark following the proof of Theorem “boundedness of semistable sheaves with fixed topology,” Section 6