2019 Fiscal Year Annual Research Report
| Project/Area Number |
17F17804
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| Research Institution | Kyoto University |
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Principal Investigator |
吉川 謙一 京都大学, 理学研究科, 教授 (20242810)
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| Co-Investigator(Kenkyū-buntansha) |
ZHANG YEPING 京都大学, 理学(系)研究科(研究院), 外国人特別研究員
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| Project Period (FY) |
2017-11-10 – 2020-03-31
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| Keywords | Analytic torsion / BCOV invariant / Calabi-Yau manifolds / birational maps |
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| Outline of Annual Research Achievements |
The BCOV invariant is an invariant for Calabi-Yau manifolds. It was conjectured that the BCOV invariant is a birational invariant. The co-investigator's ultimate goal is to prove this conjecture. The approach is based on the following theorem: any birational equivalence between algebraic varieties can be decomposed into blow-up/blow-down with smooth center. The research program consists of two steps.
1. We construct BCOV invariant for Kaehler manifolds equipped with a simple normal crossing canonical divisor, which is not necessarily effective.
2. We study the behavior of the extended BCOV invariant under blow-up.
The co-investigator accomplished step 1 and showed that for rigid del Pezzo surfaces, the invariant obtained is equivalent to Yoshikawa's equivariant BCOV invariant. The co-investigator obtained considerable progress on step 2: the change of BCOV invariant under blow-up is uniquely determined by the topological type of the center of the blow-up. This result implies a weak version of the conjecture: the BCOV invariant is a birational invariant modulo a deformation type invariant. The co-investigator also showed that the conjecture holds for Mukai flops. In the research mentioned above, the co-investigator essentially used results from birational geometry and the theory of Quillen metrics. The co-investigator expects that his research could activate new interaction between these research areas, which belong to very different branches of mathematics.
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| Research Progress Status |
令和元年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和元年度が最終年度であるため、記入しない。
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