Studying the degeneration of canonical Kähler metrics is one of the central topics in modern complex geometry. On the other hand, curvature is one of the most important quantities in Riemannian geometry. For complex surfaces, it is known that the L2-energy of the total curvature tensor converges to a linear combination of the L2-energy of the limit space (an orbifold) and Dirac masses supported on each singularity (cf. Bando 1990). In this talk, we will discuss the regularity of this convergence. In particular, we show that it is Hölder continuous in a certain sense. This talk is based on the speaker's paper (arXiv:2505.01773), which is based on his master thesis.