Analytic torsion forms for fibrations by projective curves (Q6060824)

Analytic torsion forms, which have been constructed by \textit{J.-M. Bismut} and the author in [J. Algebr. Geom. 1, No. 4, 647--684 (1992; Zbl 0784.32023)], are differential forms on the base \(B\) associated to Hermitian holomorphic vector bundles \(\overline{E}\) over fibrations \( \pi: M\to B\) of complex manifolds equipped with a certain Kähler structure. While their degree 0 part equals Ray-Singer's complex analytic torsion, there are only few explicitly known values of analytic torsion forms in higher degree. In this paper the author gives an explicit formula for analytic torsion forms for fibrations by projective curves. He also uses this to obtain a formula for direct images in Arakelov geometry. The proof mainly uses a new description of Bismut's equivariant Bott-Chern current in the case of isolated fixed points.