Algebraic limits of Hermitian-Yang-Mills connections on Kähler surfaces

We develop a geometric way to uniformly carry out Hörmander’s $L^2$-estimates for Hermitian-Yang-Mills connections with bounded energy on Kähler surfaces. We then show the Uhlenbeck convergence of smooth HYM connections on Kähler surfaces can be realized as algebraic convergence inside an analytic flat family. As applications, we prove a continuity theorem for smoothable families and a boundedness result for $\mu$-semistable sheaves with fixed topology. We also discuss potential applications for the moduli problem.