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Searching 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, employing pluripotential theory and a variational approach in families. We shall start by reviewing fundamental concepts and important properties within the variational picture of constant scalar curvature Kähler (cscK) metrics. I will then introduce notions of weak and strong topologies of quasi-plurisubharmonic functions in families, and explain several properties of entropy and Mabuchi functional extended to the family framework. Finally, I will demonstrate how these properties contribute to obtaining the stability of coercivity of the Mabuchi functional for the family parameter and the construction of cscK metrics on smoothable varieties. This talk is based on joint work with T. D. Tô and A. Trusiani.
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Singular cscK metrics on smoothable varieties
