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This article finds a structure of singular sets on compact Kahler surfaces,which Taubes introduced in the studies of the asymptotic analysis of solutionsto the Kapustin-Witten equations and the Vafa-Witten ones originally on smoothfour-manifolds. These equations can be seen as real four-dimensional analoguesof the Hitchin equations on Riemann surfaces, and one of common obstacles to beovercome is a certain unboundedness of solutions to these equations, especiallyof the "Higgs fields". The singular sets by Taubes describe part of thelimiting behaviour of a sequence of solutions with this unboundedness property,and Taubes proved that the real two-dimensional Haussdorff measures of thesesingular sets are finite. In this article, we look into the singular sets, whenthe underlying manifold is a compact Kahler surface, and find out that theyhave the structure of an analytic subvariety in this case.

On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact K\"ahler surfaces