Direct gluing of local analytic moduli structures on Kähler surfaces

Determine whether the local analytic structures associated with smoothable semistable sheaves on a compact Kähler surface can be glued by a direct method, rather than by extending the existing construction to the analytic setting.

Background

The paper outlines a three-step strategy for constructing a proper analytic moduli space of smoothable semistable sheaves and relating it to the Donaldson–Uhlenbeck compactification. Step 2 concerns globalizing local analytic structures obtained from versal deformation spaces.

The proposed approach is to extend the fundamental gluing work of AFS17 to the analytic category. The authors explicitly leave unresolved whether a more direct gluing construction is possible when the base is merely a surface.