ncts

Searching for canonical metrics in Kähler classes has been a central theme in Kähler geometry for decades. This talk aims to explain a method for investigating canonical metrics in families of singular varieties with a relative pluripotential theory and a variational approach in families. We shall start by revisiting fundamental concepts of canonical metrics and important properties within the variational picture for constant scalar curvature Kähler (cscK) metrics. I will then introduce notions of weak and strong topology of quasi-plurisubharmonic functions in families and explain several properties of entropy and Mabuchi functional extended to the family framework. Finally, I will present how these properties contribute to achieving uniform coercivity with almost optimal slopes of the Mabuchi functional and construct singular cscK potentials on smoothable varieties. This talk is based on joint work with T. D. Tô and A. Trusiani.