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Complex analytic torsion forms for torus fibrations and moduli spaces. (English) Zbl 1071.58026

Reznikov, Alexander (ed.) et al., Regulators in analysis, geometry and number theory. Basel: Birkhäuser (ISBN 0-8176-4115-7/hbk). Prog. Math. 171, 167-195 (2000).
The main purpose of this paper is to construct analytic torsion forms for torus fibrations which do not need to be Kähler fibrations. This extends work of J.-M. Bismut, H. Gillet and C. Soulé [Commun. Math. Phys. 115, No. 1, 79–126 (1988; Zbl 0651.32017)], H. Gillet, C. Soulé [Topology 30, No. 1, 21–54 (1991; Zbl 0787.14005)], G. Faltings [Lectures on the arithmetic Riemann-Roch theorem, Ann. Math. Stud., 127, Princeton Univ. Press, (Princeton, NJ), (1992; Zbl 0744.14016)], and J.-M. Bismut and K. Köhler [J. Algebr. Geom. 1, No. 4, 647–684 (1992; Zbl 0784.32023)] for special Kähler fibrations. Such torsion forms are the main ingredient of a direct image construction for an Hermitian \(K\)-theory, which was developed by H. Gillet, C. Soulé in the context of Arakelov theory [Ann. Math. (2) 131, 163–203 (1990; Zbl 0715.14018), Ann. Math. (2) 131, No. 2, 205–238 (1990; Zbl 0715.14006)].
More precisely, the author considers the following setting. Let \(\pi:(E^{1,0},g^E)\to B\) be an \(n\)-dimensional holomorphic Hermitian vector bundle on a compact complex manifold, and let \(\Lambda\) be a lattice, spanning the underlying real bundle \(E\) of \(E^{1,0}\), so that local sections of \(\Lambda\) are holomorphic sections of \(E^{1,0}\). Then \(\pi: E^{1,0}/\Lambda^{1,0}\to B\) is a holomorphic torus fibration which is not necessarily flat as a complex fibration, and the corresponding analytic torsion form is constructed by explicitly double transgressing the top Chern class of \(E^{0,1}\), which was proved to be \(0\) in cohomology by D. Sullivan [C. R. Acad. Sci., Paris, Sér. A 281, 17–18 (1975; Zbl 0312.55022)].
For the entire collection see [Zbl 1038.11002].

MSC:

58J52 Determinants and determinant bundles, analytic torsion
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
32L05 Holomorphic bundles and generalizations
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