Analyticity of the Donaldson–Uhlenbeck compactification for compact Kähler surfaces

Establish that the Donaldson–Uhlenbeck compactification of the moduli space of smooth irreducible Hermitian–Yang–Mills connections on a general compact Kähler surface carries a complex analytic structure extending the natural complex analytic structure on the smooth locus, without assuming that the surface or its Kähler class is algebraic.

Background

For compact Kähler surfaces, the paper constructs algebraic limits of sequences of Hermitian–Yang–Mills connections inside analytic flat families and derives continuity and boundedness results for semistable sheaves. These results provide local analytic parameterizations near the Donaldson–Uhlenbeck boundary but do not construct a global analytic structure on the entire compactified moduli space.

The unresolved problem is motivated by known results in the integrally polarized projective case and by the expectation that analogous analyticity should hold for arbitrary compact Kähler surfaces, including nonalgebraic surfaces and nonalgebraic Kähler classes. The paper explicitly presents this as a folklore conjecture that remains unresolved.