K\"ahler-Einstein metrics with prescribed singularities on Fano manifolds

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