  It is well known that degree two Deligne cohomology groups can be identified
with groups of isomorphism classes of holomorphic line bundles with
connections. There is also a geometric description of degree three Deligne
cohomology, due to J-L. Brylinski and P. Deligne, in terms of gerbes with
connective structures and curvings. This paper gives a geometric interpretation
of Deligne cohomology of all degrees, in terms of equivalence classes of higher
line bundles with $k$-connections. It is also shown that the classical
Abel-Jacobi isomorphism $Pic^0(X) \cong J(X)$ generalizes to the isomorphism
between groups of equivalence classes of topologically trivial 1-holomorphic
higher line bundles with $k$-connections and Griffiths intermediate Jacobians.
