The existence of constant scalar curvature Kähler (cscK) metrics is a fundamental problem in complex geometry. In this talk, we study this problem in the case when cscK metrics have singularities. In particular, we consider two types of singularities: Poincaré type singularities and conical singularities. Firstly, I will explain that a cscK metric of Poincaré type can be approximated by cscK cone metrics under some assumption. Secondly, I will explain that the existence of a cscK metric of Poincaré type implies log K-semistability for corresponding cone angle.
This result is related to conjectures by J.Sun-S.Sun and G.Székelyhidi.