2017 Fiscal Year Research-status Report
| Project/Area Number |
16K17558
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| Research Institution | The University of Tokyo |
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Principal Investigator |
江 辰 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (90772773)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | Fano varieties / boundedness / K-stability / alpha-invariants / pluri-canonical system |
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| Outline of Annual Research Achievements |
In FY2017, I continued to study the boundedness of Fano varieties.As the Borisov-Alexeev-Borisov Conjecture was solved by Birkar last year, I participated a workshop at NCTS, Taiwan on April to explain his work, and we are planning to write a book to explain the ideas of his work on BAB Conjecture.Then I turned to study boundedness of K-semistable Fano varieties. Both K-stability and boundedness are central topics of the study of Fano varieties. K-stability is expected to be the right condition in order to construct a good moduli space for Fano varieties and boundedness is the first step towards the construction of a moduli. So this study combines these two central topics. On May, I proved that K-semistable Fano varieties with volumes bounded from below form a bounded family.Later, I continued my joint work with Meng Chen at Fudan University on the study of the explicit geometry of terminal (weak) Fano 3-folds. We studied the behavior of pluri-canonical system of terminal weak Fano 3-folds, and showed that any terminal weak Fano 3-fold is birational to another terminal weak Fano 3-fold with the 52nd-canonical system giving a birational map. This result generalized our first joint work which is published on JDG in 2016.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
This year I proved the boundedness of K-semistable Fano varieties with bounded volumes. This progress is somehow unexpected. The result related two central area of the study of Fano varieties and might be important for constructing moduli.
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| Strategy for Future Research Activity |
In the future, I am going to continue the study of boundedness of Fano varieties and related topics. Note that I was planning to study the BAB Conjecture when I start this research project, but now it has been solved by Birkar. However, boundedness of Fano varieties is a very deep and interesting area and still has many open problems. I will mainly concentrate on the following topics:
1.Constructing good moduli for certain Fano varieties. As I showed the boundedness of K-semistable Fano varieties with anti-canonical degrees bounded from below, it is natural to consider the construction of moduli.
2.Boundedness in positive characteristics. As minimal model program was developed in dimension 3 in characteristic p>5, it is interesting to ask what we can say about boundedness of singular Fano 3-folds in characteristic p>5.
3.Boundedness of rationally connected Calabi-Yau varieties. Boundedness of Calabi-Yau varieties is a more challenging problem. As rationally connected Calabi-Yau varieties behave very like Fano varieties, we may expect to show the boundedness of rationally connected Calabi-Yau varieties by method in the study of Fano varieties.
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