Geometry of Deligne cohomology

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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 Pic⁰(X) 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.