论文标题
终止翻转和异常非规范的奇异点
On termination of flips and exceptionally non-canonical singularities
论文作者
论文摘要
我们系统地介绍并研究了一种新型的奇异性，即异常非典型的（ENC）奇点。这类奇异性在研究生物几何学的许多问题中起着重要作用，并且与局部K稳定性理论，Calabi-yau品种和镜面对称性具有紧密的联系。 我们将翻转的终止终止终止倒入终端翻转和ACC猜想的终止，以减少ENC对的最小对数差异（MLDS）。结果，ENC对MLDS的ACC猜想意味着在尺寸$ 4 $中的翻转终止。 我们表明，在任何固定的维度上，翻转的终止均来自终端对的MLDS的低血管内，以及末端和ENC对的MLDS的ACC。此外，在尺寸$ 3 $中，我们对ENC奇点进行了粗略的分类，并证明了ENC对MLD的ACC。这两个结果提供了第二个证明，限制$ 3 $的翻转终止，这不依赖任何困难功能。 最后，我们提出并证明了关于ENC奇异性和局部K稳定性理论的几种猜想的特殊情况。我们还通过镜像对称性讨论了奇异的奇异品，特殊的Fano品种和带有小型MLD的Calabi-Yau品种之间的关系。
We systematically introduce and study a new type of singularities, namely, exceptionally non-canonical (enc) singularities. This class of singularities plays an important role in the study of many questions in birational geometry, and has tight connections with local K-stability theory, Calabi-Yau varieties, and mirror symmetry. We reduce the termination of flips to the termination of terminal flips and the ACC conjecture for minimal log discrepancies (mlds) of enc pairs. As a consequence, the ACC conjecture for mlds of enc pairs implies the termination of flips in dimension $4$. We show that, in any fixed dimension, the termination of flips follows from the lower-semicontinuity for mlds of terminal pairs, and the ACC for mlds of terminal and enc pairs. Moreover, in dimension $3$, we give a rough classification of enc singularities, and prove the ACC for mlds of enc pairs. These two results provide a second proof of the termination of flips in dimension $3$ which does not rely on any difficulty function. Finally, we propose and prove the special cases of several conjectures on enc singularities and local K-stability theory. We also discuss the relationship between enc singularities, exceptional Fano varieties, and Calabi-Yau varieties with small mlds or large indices via mirror symmetry.
