  We develop a theory of Cech-Bott-Chern cohomology and in this context we
naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern
cohomology has two relatives and they all arise from a single complex. Thus we
study these three cohomologies in a unified way and obtain a long exact
sequence involving the three. We then study the localization problem of
characteristic classes in the relative Bott-Chern cohomology. For this we
define the cup product and integration in our framework and we discuss local
and global duality homomorphisms. After reviewing some materials on
connections, we give a vanishing theorem relevant to our localization. With
these, we prove a residue theorem for a vector bundle admitting a Hermitian
connection compatible with an action of the non-singular part of a singular
distribution. As a typical case, we discuss the action of a distribution on the
normal bundle of an invariant submanifold (so-called the Camacho-Sad action)
and give a specific example.
