Transversal holomorphic sections and localization of analytic torsions
The paper concerns localization of certain invariants related to complex manifolds and the main result of it is three-fold: First, it is a Bott-type localization formula relating the integral of certain forms (twisted by a holomorphic bundle) on a complex manifold over this entire manifold to the integrals over the zero set (complex submanifold) of a transversal holomorphic section of the mentioned bundle. Second, the authors give a formula for the Quillen metric of an isomorphism between two complex lines (determinants) related to the situation described above [as defined e.g., by \textit{J. M. Bismut} and \textit{G. Lebeau}, Publ. Math., Inst. Hautes Etudes Sci. 74, 1--297 (1991; Zbl 0784.32010)] expressing this metric in terms of characteristic numbers of the submanifold and the manifold itself. Third, the authors provide a formula for certain analytic torsions of the entire manifold to the analytic torsions of the zero submanifold of the holomorphic section. The formulas generalize some earlier results of Bismut, Lebeau, Zhang and the authors of the present paper, and are too complicated to be cited here. The paper is not self contained and familiarity with the earlier papers cited in the bibliography is necessary to grasp the essence of it.
scientific article; zbMATH DE number 5011483