an:00247024
Zbl 0784.32023
Bismut, Jean-Michel; K??hler, Kai
Higher analytic torsion forms for direct images and anomaly formulas
EN
J. Algebr. Geom. 1, No. 4, 647-684 (1992).
00014153
1992
j
32L05 14G40 53C05
analytic torsion forms; K??hler fibrations; anomaly; superconnections
The authors construct analytic torsion forms associated to K??hler fibrations and establish corresponding anomaly formulas. In \S1, one recalls results concerning the Levi-Civita superconnections and K??hler fibrations. In \S2, one proves variation formulas for the Levi-Civita superconnection and the corresponding heat kernel supertraces in terms of the (1,1)-form \(\omega\) of the K??hler fibration. In \S3, one constructs analytic torsion forms \(T(\omega,h^ \xi)\) associated to K??hler fibrations for nonacyclic complexes, whose cohomology groups form a vector bundle on the base. The main result, Theorem 3.10, describes the dependence of \(T(\omega,h^ \xi)\) on \(\omega\) and \(h^ \xi\). As a corollary, one proves in Theorem 3.11 that the class of \(T(\omega,h^ \xi)\) (modulo \(\partial\) and \(\overline\partial\) coboundaries) only depends on the natural holomorphic and metric datas of the problem. These anomaly formulas make these analytic torsion forms ``natural'' in Arakelov arithmetic geometry.
Vasile Br??nz??nescu (Bucure??ti)