  The main objective of the present paper is to set up the theoretical basis
and the language needed to deal with the problem of direct images of hermitian
vector bundles for projective non-necessarily smooth morphisms. To this end, we
first define hermitian structures on the objects of the bounded derived
category of coherent sheaves on a smooth complex variety. Secondly we extend
the theory of Bott-Chern classes to these hermitian structures. Finally we
introduce the category $\oSm_{\ast/\CC}$ whose morphisms are projective
morphisms with a hermitian structure on the relative tangent complex.
