2017 Fiscal Year Annual Research Report
Project/Area Number 
17F17804

Research Institution  Kyoto University 
Principal Investigator 
吉川 謙一 京都大学, 理学研究科, 教授 (20242810)

CoInvestigator(Kenkyūbuntansha) 
ZHANG YEPING 京都大学, 理学(系)研究科(研究院), 外国人特別研究員

Project Period (FY) 
20171110 – 20200331

Keywords  analytic torsion / scattering theory 
Outline of Annual Research Achievements 
The RaySinger torsion (RStorsion) is a spectral invariant of manifolds. Its topological counterpart is known as Reidemeister torsion (Rtorsion). The relation between RStorsion and Rtorsion are clarified by the works of Cheeger, Mueller, Bismut and W. Zhang (CMBZTheorem). Using CMBZTheorem, Bruening and Ma proved a gluing formula for RStorsion. Both the RStorsion and Rtorsion were generalized to families of manifolds, known as BismutLott analytic torsion form (BLtorsion) and IgusaKlein higher torsion (IK torsion). However, the family version of CMBZTheorem 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 BLtorsion verifies the transfer axiom and that IKtorsion verifies both axioms. In a joint work with Martin Puchol and Jialin Zhu, the coinvestigator established a gluing formula for BLtorsion, 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 BLtorsion verifies the additivity axiom. As a consequence, BLtorsion and IKtorsion are equivalent up to a universal class.

Current Status of Research Progress 
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The result obtained is exactly what the coinvestigator proposed in his research plan submitted in the last year.

Strategy for Future Research Activity 
The Quillen metric is a spectral invariant of Kaehler manifolds. In certain cases, the Quillen metric is independent of the Kaehler form, e.g., CalabiYau manifolds. The behavior of Quillen metrics under submersion/immersion was wellstudied. However the behavior of Quillen metrics under blowup remains mysterious expect for several special cases. The coinvestigator attempts to understand the behavior of Quillen metrics under blowup 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 CalabiYau orbifold.
