  The Chern classes of a K-theory class which is represented by a vector bundle
with connection admit refinements to Cheeger-Simons classes in Deligne
cohomology. In the present paper we consider similar refinements in the case
where the classes in K-theory are represented by geometric families of Dirac
operators. In low dimensions these refinements correspond to the exponentiated
eta-invariant, the determinant line bundle with Quillen metric and Bismut-Freed
connection, and Lott's index gerbe with connection and curving. We give a
unified treatement of these cases as well as their higher generalizations.
  Our main technical tool is a variant of local index theory for Dirac
operators of families of manifolds with corners.
