| 研究実績の概要 |
The Ray-Singer torsion (RS-torsion) is a spectral invariant of manifolds. Its topological counterpart is known as Reidemeister torsion (R-torsion). The relation between RS-torsion and R-torsion are clarified by the works of Cheeger, Mueller, Bismut and W. Zhang (CMBZ-Theorem). Using CMBZ-Theorem, Bruening and Ma proved a gluing formula for RS-torsion.
Both the RS-torsion and R-torsion were generalized to families of manifolds, known as Bismut-Lott analytic torsion form (BL-torsion) and Igusa-Klein higher torsion (IK- torsion). However, the family version of CMBZ-Theorem remained unknown. Igusa axiomatized the higher torsion invariants. His axiomatization consists of additivity axiom and transfer axiom. Two higher torsion invariants verifying the axioms are equivalent up to a universal class. It is known that BL-torsion verifies the transfer axiom and that IK-torsion verifies both axioms.
In a joint work with Martin Puchol and Jialin Zhu, the co-investigator established a gluing formula for BL-torsion, which extends the result of Bruening and Ma. The proof is based on the adiabatic limit, which makes their proof very different from the work of Bruening and Ma.
Their formula shows that BL-torsion verifies the additivity axiom. As a consequence, BL-torsion and IK-torsion are equivalent up to a universal class.
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| 今後の研究の推進方策 |
The Quillen metric is a spectral invariant of Kaehler manifolds. In certain cases, the Quillen metric is independent of the Kaehler form, e.g., Calabi-Yau manifolds. The behavior of Quillen metrics under submersion/immersion was well-studied. However the behavior of Quillen metrics under blow-up remains mysterious expect for several special cases.
The co-investigator attempts to understand the behavior of Quillen metrics under blow-up in the general case. The argument will be based on the adiabatic limit. The result expected may lead to several interesting consequences, e.g., to calculate the BCOV invariant for a crepant resolution of a Calabi-Yau orbifold.
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