Proceedings of the London Mathematical Society Advance Access originally published online on February 14, 2008
Proceedings of the London Mathematical Society 2008 97(1):1-30; doi:10.1112/plms/pdm039
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© 2008 London Mathematical Society
D-bar sparks
Mathematics Department
Rice University
Houston, TX 77005-1892
USA
Mathematics Department
Stony Brook University
Stony Brook, NY 11794-3651
USA
blaine@math.sunysb.edu
Received 27 July 2006. Revision received 27 July 2007.
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 
x-gerbes with connection, it yields a soft resolution of the sheaf x 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 
*(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.
The research is partially supported by the NSF.
2000 Mathematics Subject Classification 14F43, 49Q15, 58A25.