| 研究実績の概要 |
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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| 今後の研究の推進方策 |
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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