Generalized holomorphic analytic torsion

Abstract

In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and K¨ohler is obtained as the theory of generalized analytic torsion classes associated to $–R=2$, $R$ being the $R$-genus. As an application of the axiomatic characterization, we give new simpler proofs of known properties of holomorpic analytic torsion classes, we give a characterization of the $R$-genus, and we construct a direct image of hermitian structures for projective morphisms.