Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs
Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs
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DOI：
10.1007/s00039-014-0301-8
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发表时间：
2013-04
影响因子：
2.2
通讯作者：
R. Berman;Henri Guenancia
中科院分区：
文献类型：
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作者：
R. Berman;Henri Guenancia
设X是一个标准极化簇，即一个复射影簇，使得它的标准类KX定义一个样本线丛，并且满足条件G1和S2。我们的主要结果是，如果X具有半对数正则奇异性，则X承认凯勒-爱因斯坦度量，即如果X是Kollár-Shepherd-巴伦和Alexeev意义上的稳定簇（已知其模空间是紧的）。根据定义，在这个奇异的上下文中，凯勒-爱因斯坦度规仅仅意味着在X的正则轨迹上的凯勒-爱因斯坦度规，其体积等于KX的代数体积，即KX的顶部相交数。我们还表明，这样的度量是唯一确定的，并扩展到定义一个典型的正电流inc 1（KX）。结合Odaka的最新结果，我们的主要结果表明X允许Kähler-Einstein度规且X是K-稳定的，从而证实了Yau-Tian-唐纳森猜想在（可能是奇异的）正则极化簇的一般情况下成立.更一般地说，我们的结果被证明是在设置的日志最小品种，他们也推广了一些以前的结果关于Kähler-Einstein度量的拟投射品种。
LetXbe a canonically polarized variety, i.e. a complex projective variety such that its canonical classKXdefines an ample-line bundle, and satisfying the conditionsG1andS2. Our main result says thatXadmits a Kähler–Einstein metric iffXhas semi-log canonical singularities i.e. iffXis a stable variety in the sense of Kollár–Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler–Einstein metric in this singular context simply means a Kähler–Einstein on the regular locus ofXwith volume equal to the algebraic volume ofKX, i.e. the top intersection number ofKX. We also show that such a metric is uniquely determined and extends to define a canonical positive current inc1(KX). Combined with recent results of Odaka our main result shows thatXadmits a Kähler–Einstein metric iffXisK-stable, which thus confirms the Yau–Tian–Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kähler–Einstein metrics on quasi-projective varieties.