Richard B. Melrose and Paolo Piazza
For a family of Dirac operators, acting on Hermitian Clifford modules over
the odd-dimensional compact manifolds with boundary which are the fibres of a
fibration with compact base, we compute the Chern character of the index, in
of the base. Although we assume a product decomposition near the
boundary we make no assumptions on invertibility of the boundary family and
instead obtain a family of self-adjoint Fredholm operators by choice of an
auxiliary family of projections respecting the 
decomposition of
bundles over the boundary. In case the boundary family is invertible this
projection can be taken to be the Atiyah-Patodi-Singer projection and the
resulting formula is as conjectured by Bismut and Cheeger. The derivation
of the index formula is effected by the combination of the superconnection
formalism of Quillen and Bismut, the calculus of b-pseudodifferential
operators and suspension.