Abstract:In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a Kähler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold which is complete towards the vertex. This provides a local model for the further study of the global Kähler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is the desired one. In the second approach, we obtain a Kähler-Einstein metric with the desired properties by Calabi Ansatz. At last, we show that the obtained metrics are the same.

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中文翻译：

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关于在孤立对数正则奇点附近具有负标量曲率的完整 Kähler-Einstein 度量的构建

摘要：在这篇简短的笔记中，我们关注锥型对数正则奇点附近的 Kähler-Einstein 度量。通过两种不同的方法，我们在朝向顶点是完整的 Calabi-Yau 流形上的锥体的邻域中构造了具有负标量曲率的 Kähler-Einstein 度量。这为进一步研究关于奇异变体的全球 Kähler-Einstein 度量提供了一个局部模型。在第一种方法中，我们表明奇点被复数球均匀化，因此来自球的 Bergman 度量的诱导度量是所需的。在第二种方法中，我们通过 Calabi Ansatz 获得具有所需属性的 Kähler-Einstein 度量。最后，我们表明获得的指标是相同的。

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