Kähler-Einstein metrics with prescribed singularities on Fano manifolds

Abstract

Given a Fano manifold $(X,\omega )$ 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 ${\alpha }_{\omega }$ 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 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 $\alpha$-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.

arXiv commentDefinition of the ${\alpha }_{\omega }$-function and Theorem A modified, other related changes. Improved and final version: to appear in "Journal für die reine und angewandte Mathematik (Crelle's Journal)"

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Cited by in corpus (2)

• Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting
• Geodesics in the space of $m$-subharmonic functions with bounded energy