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

Abstract

This article finds a structure of singular sets on compact Kahler 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 Kahler surface, and find out that they have the structure of an analytic subvariety in this case.