Singular cscK metrics on smoothable varieties

Abstract

We prove the lower semi-continuity of the coercivity threshold of Mabuchi functional along a degenerate family of normal compact Kähler varieties with klt singularities. Moreover, we establish the existence of singular cscK metrics on $\mathbb{Q}$-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.

47 pages; v2: removed appendix, changed the proof of Theorem 5.4, corrected some typos; v3: minor modifications to the proofs of Theorem 3.5 and Proposition 5.6, Section 5.4 added