Abstract
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.
Highlights
Two sets of gauge-theoretic equations that we consider in this article originated in N = 4 super Yang–Mills theory in theoretical physics; they appear after topological twists of the theory
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”
Summary
Taubes introduces some real codimension two singular set, to be denoted Z , outside which a sequence of “partially rescaled” S L(2, C)-connections converges after gauge transformations except bubbling out at a finite set of points (see [2,4,9,15,16,17,21] and [22] for related problems) Vafa–Witten equations case, namely, the singular set can be identified with the zero set of a section of the square of the canonical bundle of X , so it has a structure of an analytic subvariety of X (Corollary 3.5). We consider the equations on compact Kähler surfaces, and prove a similar statement to Theorem 1.2 for the Vafa–Witten case in Sect.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
