Kähler-Einstein metrics with prescribed singularities on Fano manifolds

Abstract

Given a Fano manifold $(X,ω)$ we develop a variational approach to characterize analytically the existence of Kähler-Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function $α_ω$ on the set of prescribed singularities which generalizes Tian's $α$-invariant, showing that its upper level set $\{α_ω(\cdot)>\frac{n}{n+1}\}$ produces a subset of the Kähler-Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler-Einstein metrics. In particular, we prove that many $K$-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the $α$-invariant function at non-trivial prescribed singularities (or other conditions) implies the existence of genuine Kähler-Einstein metrics. Finally, through a continuity method, we also prove the strong continuity of Kähler-Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.