Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Given a Fano manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>α</m:mi> <m:mi>ω</m:mi> </m:msub> </m:math> {αω} on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>α</m:mi> <m:mi>ω</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>></m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfrac> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{αω(\,·\,)>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 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.