Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Given a Fano manifold .X; !/, 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! on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set 1α!. / > nCn1ºproduces 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 nontrivial 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.