D-bar Sparks, I

F. Reese Harvey; H. Blaine Lawson Jr

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arXiv · math/0512247

D-bar Sparks, I

F. Reese Harvey·H. Blaine Lawson Jr

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Abstract

A d-bar-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 d-bar-spark classes H^*(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.

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F. Reese Harvey, H. Blaine Lawson Jr. 2005-12-12. D-bar Sparks, I. https://arxiv.org/abs/math/0512247

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