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

DOI10.1515/CRELLE-2022-0047arXiv2006.09130MaRDI QIDQ6343031

Antonio Trusiani

Publication date: 16 June 2020

Abstract: 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 $alph{a}_{omega}$ on the set of prescribed singularities which generalizes Tian's $alpha$-invariant, showing that its upper level set $alph{a}_{omega}(cdot)>fracnn+1$ 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.

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Fano varieties (14J45) Kähler-Einstein manifolds (32Q20)

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