On termination of flips and exceptionally non-canonical singularities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913667158638592 |
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| author | Han, Jingjun Liu, Jihao |
| author_facet | Han, Jingjun Liu, Jihao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_13122 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On termination of flips and exceptionally non-canonical singularities Han, Jingjun Liu, Jihao Algebraic Geometry 14E30, 14B05, 32S05, 14J17, 14J30, 14J35 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. |
| title | On termination of flips and exceptionally non-canonical singularities |
| topic | Algebraic Geometry 14E30, 14B05, 32S05, 14J17, 14J30, 14J35 |
| url | https://arxiv.org/abs/2209.13122 |