AI 中文总结
本文通过对数极小模型程序将 Song-Tian 程序应用于轻度奇异对，证明了存在唯一的纤维化锥形 Kähler-Einstein 度量，并将完整答案归结为纤维化 Calabi-Yau 叶状结构的 CMA 方程。
AI 中文摘要
在射影奇异簇的典范模型上存在典范度量是一个长期存在的猜想，该猜想的主要部分涉及没有确定第一 Chern 类的簇（大多数簇没有确定第一 Chern 类）。有一个被称为 Song-Tian 程序的方案，旨在通过使用极小模型程序在射影簇的典范模型上寻找典范度量。在本文中，我们通过对数极小模型程序将 Song-Tian 程序应用于轻度奇异对 $(X;D)$，其中 $D$ 是 $X$ 上具有锥奇点的简单正规交叉除子。我们证明，在 $(X;D)$ 上存在唯一的 $C^\infty$-纤维化锥形 Kähler-Einstein 度量，其 Lelong 数为零，该度量由对数 Weil-Petersson 度量和 Fujino-Mori [72] 的附加项扭曲，只要我们具有纤维化 KE-稳定性或 Kawamata 的定理 2.28 或 2.30（在 $C^0$ 情形）的条件。最后，我们强调如何将这一问题的完整答案归结为对应于纤维化 Calabi-Yau 叶状结构（归功于 H.Tsuji）的 CMA 方程，该方程仍然开放。
英文摘要
Existence of canonical metric on a canonical model of projective singular variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class (most of the varieties do not have definite first Chern class). There is a program which is known as Song-Tian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. In this paper, we apply Song-Tian program for mildly singular pair $(X;D)$ via Log Minimal Model Program where $D$ is a simple normal crossing divisor on $X$ with conic singularities. We show that there is a unique $C^\infty$-fiberwise conical Kähler-Einstein metric on $(X;D)$ with vanishing Lelong number which is twisted by logarithmic Weil-Petersson metric and an additional term of Fujino-Mori [72] as soon as we have fiberwise KE-stability or Kawamata's condition of Theorems 2.28, or 2.30(in $C^0$ case). In final we highlight that how the complete answer of this question can be reduced to CMA equation corresponding to fiberwise Calabi-Yau foliation(due to H.Tsuji) which is still open.
