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Higher analytic torsion forms for direct images and anomaly formulas. (English) Zbl 0784.32023
The authors construct analytic torsion forms associated to Kähler fibrations and establish corresponding anomaly formulas. In §1, one recalls results concerning the Levi-Civita superconnections and Kähler fibrations. In §2, 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 Kähler fibration. In §3, one constructs analytic torsion forms \(T(\omega,h^ \xi)\) associated to Kähler 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.

32L05 Holomorphic bundles and generalizations
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
53C05 Connections, general theory