2018 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 |
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi- Yau manifolds, which is now called BCOV invariant.
The co-investigator extended the BCOV invariant to Calabi-Yau pairs, i.e., a Kaehler manifold equipped with a pluricanonical divisor. The co-investigator proved that the extended BCOV invariant for rigid del Pezzo surfaces is equivalent to Yoshikawa’s equivariant BCOV invariant. The co-investigator also explored the behavior of the extended BCOV invariant under blowing up. More precisely, the co-investigator showed that the difference between the BCOV invariants of the initial manifold and its blowing up, viewed as a function on the moduli space (of the manifold in question and the center of the blowing up), is pluriharmonic. The co-investigator also obtained an explicit formula in the special case where the manifold in question is of dimension two.
The results obtained may serve as an intermediate step towards Yoshikawa’s conjecture that the BCOV invariant for Calabi-Yau threefolds is a birational invariant. More precisely, a birational equivalence between two Calabi-Yau threefolds may be decomposed into a series of blowing up/down. Hence the problem is decomposed into each blowing up/down.
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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
The progress in the academic year 2018 is unexpected. The results obtained indicate a new approach to Yoshikawa’s conjecture that the BCOV invariant for Calabi-Yau threefolds is a birational invariant. Recall that we proposed a different approach to the conjecture mentioned above in the research plan for 2018. Comparing with the old approach, the new one seems to be easier and more feasible.
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| Strategy for Future Research Activity |
The research plan consists of two independent topics.
Topic I. Bismut, Ma and Zhang considered a double fibration and studied the asymptotics of the analytic torsion forms of the lower fibration equipped with the direct image of the upper fibration. The co-investigator showed that the analytic torsion form considered by Bismut, Ma and Zhang is the adiabatic limit of the analytic torsion forms of the total fibration. The co-investigator attempts to understand the asymptotics of the analytic torsion forms of the total fibration.
Topic II. This the the continuation of the research in the academic year 2018. The co-investigator attempts to extend the results obtained to more general cases. The ultimate goal is to prove Yoshikawa’s conjecture (stated in the summary of research achievements).
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