D-bar sparks

A ∂-analog of differential characters for complex manifolds is introduced and studied using a new theory of homological spark complexes. Many essentially different spark complexes are shown to have isomorphic groups of spark classes. This has many consequences: it leads to an analytic representation of ^×-gerbes with connection, it yields a soft resolution of the sheaf ^× by currents on the manifold and, more generally, it gives a Dolbeault-Federer representation of Deligne cohomology as the cohomology of certain complexes of currents. It is shown that the ∂-spark classes Ĥ*(X) carry a functorial ring structure. Holomorphic bundles have Chern classes in this theory, which refine the integral classes and satisfy Whitney duality. A version of Bott vanishing for holomorphic foliations is proved in this context.