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We study the following question: Let(  X  ,  g  )be a compact Gauduchon surface,(  E  ,  h  )be a differentiable rank    r    vector bundle on    X   ,    D    be a fixed holomorphic structure onD  ≔  det   (  E  )and    a    be the Chern connection of the pair(  D  ,  det   (  h  )   )    . Does the complex space structure onM    a    ASD(  E  )*induced by the Kobayashi–Hitchin correspondence extend to a complex space structure on the Donaldson–Uhlenbeck compactificationM    ¯a    ASD(  E  )? Our results answer this question in detail for the moduli spaces ofSU   (  2  )-instantons withc    2    =  1on general (possibly unknown) class VII surfaces.

On the Donaldson–Uhlenbeck compactification of instanton moduli spaces on class VII surfaces