论文标题
Kähler-Einstein指标，有规定的奇异性在Fano歧管上
Kähler-Einstein metrics with prescribed singularities on Fano manifolds
论文作者
论文摘要
考虑到Fano歧管$（x，ω）$，我们开发了一种变异方法来分析表征具有规定奇异性的Kähler-Einstein指标的存在，假设这些奇异性可以通过代数近似。此外，我们在一组规定的奇点上定义了$α_Ω$，该功能概括了Tian的$α$ invariant，这表明其上层集合$ \ {α_Ω（\ cdot）> \ frac {n} {n} {n+1} {n+1} \} $由kähller-noce of the Incein of the subirection in t offeriase i.echhler-noce in of。承认Kähler-Einstein指标。特别是，我们证明许多$ k $稳定的流形承认所有可能的Kähler-Einstein指标都有规定的奇异性。相反，我们表明，在非平凡的处方奇异点（或其他条件）上$α$ invariant功能的足够阳性意味着存在真正的Kähler-Einstein指标。最后，通过一种连续性方法，我们还证明了Kähler-Einstein指标在完全有序的处方奇异点的曲线上具有强烈的连续性，当时相对构成群体是离散的。
Given a Fano manifold $(X,ω)$ 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 $α_ω$ on the set of prescribed singularities which generalizes Tian's $α$-invariant, showing that its upper level set $\{α_ω(\cdot)>\frac{n}{n+1}\}$ 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 $α$-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.
