D-bar Sparks, I

A ∂-analogue 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 O ×-gerbes with connection, it yields a soft resolution of the sheaf O × 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 H * (X) carry a functo-rial 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 holomor-phic foliations is proved in this context.