  Using the concept of a cohesive module defined by Block, we use the theory of
superconnections in the sense of Quillen to construct natural superconnections
on Hermitian cohesive modules. By the Chern-Weil construction, we obtain
characteristic classes with values in Bott-Chern cohomology which refines the
usual deRham cohomology. The main result of this paper shows that such
characteristic classes are independent of the Hermitian metric and only
dependent on the homotopy class of the cohesive module in the homotopy
category. This extends the Bott-Chern cohomology to the bounded derived
category of complexes of analytic sheaves with coherent cohomology. We also
constructed secondary Bott-Chern characteristic forms and the main technical
theorem proves that they are well defined.
