On the singular sets of solutions to the Kapustin–Witten equations and the Vafa–Witten ones on compact Kähler surfaces

This article finds a structure of singular sets on compact Kähler surfaces, which Taubes introduced in the studies of the asymptotic analysis of solutions to the Kapustin– Witten equations and the Vafa–Witten ones originally on smooth four-manifolds. These equations can be seen as real four-dimensional analogues of the Hitchin equations on Riemann surfaces, and one of common obstacles to be overcome is a certain unboundedness of solutions to these equations, especially of the “Higgs fields”. The singular sets by Taubes describe part of the limiting behaviour of a sequence of solutions with this unboundedness property, and Taubes proved that the real two-dimensional Haussdorff measures of these singular sets are finite. In this article, we look into the singular sets,when the underlying manifold is a compact Kähler surface, and find out that they have the structure of an analytic subvariety in this case.