Abstract

We extend the notion of cscK metrics to singular varieties. We establish the existence of these canonical metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics. A key point is the lower semicontinuity of the coercivity threshold of Mabuchi functional along degenerate families of normal compact Kähler varieties with klt singularities. The latter suggests the openness of (uniform) K-stability for general polarized families of normal projective varieties. This is a joint work with Chung-Ming Pan and Tat Dat Tô.