2016 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 varieities / Boundedness / Minimal models / Mori fiber spaces |
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| Outline of Annual Research Achievements |
This project focuses on properties of Fano varieties, in particular boundedness of Fano varieties. In FY 2016, we have the following achievements:
1. Working with C.D. Hacon, we apply a boundedness result of Fano varieties recently proved by C. Birkar to solve a conjecture by Lehmann-Tanimoto-Tschinkel.
2. Working with Yalong Cao, we investigate Kawamata's effective non-vanishing conjecture for Calabi-Yau and Fano manifolds.
3. Working with Pu Cao, we classified torsion exceptional sheaves on weak del Pezzo surfaces.
4. We showed that the only K-semistable Fano manifold with the smallest alpha-invariant is the projective space.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We studied Fano varieties from different aspects and got fruitful results with might have more applications latter.
For example, the joint work with Hacon is related to the arithmetic properties of Fano varieties, and it was applied by Lehmann and Tanimoto to study the geometric properties of Fano varieties in order to investigate Manin's conjecture; the joint work with Pu Cao investigates Fano varieties in the point view of derived category; and the work on alpha-invariants is related to K-stability of Fano varieties, which is helpful in constructing moduli spaces of Fano manifolds.
In other word, by investigate different properties of Fano varieties from different points of view, we hope to get better understand of Fano varieties.
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| Strategy for Future Research Activity |
I have two plans in FY2017.
1. Study boundedness of Fano varieties and its relation with K-stability. One goal is to show that K-semistable Fano varieties with anti-canonical volumes bounding from below form a bounded family. This is confirmed in dimension two. In higher dimension it might be challenging, but very interesting.
2. Study explicit birational geometry of Fano 3-folds. In 2014, working with Meng Chen, we show that for a Q-factorical terminal Fano 3-fold with Picard number one, the 39-th anti-pluri-canonical map is birational. We also got some results for canonical Fano 3-folds, but it is not satisfactory. We will try to study deeper in this direction.
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