Annals of Global Analysis and Geometry

, Volume 53, Issue 3, pp 445–466 | Cite as

Twisted smooth Deligne cohomology

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Abstract

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.

Keywords

Deligne cohomology Differential cohomology Twisted cohomology Local coefficient systems Connections Čech-de Rham complex Smooth stacks 

Notes

Acknowledgements

The authors would like to thank the organizers and participants of the Geometric Analysis and Topology Seminar at the Courant Institute for Mathematical Sciences for asking about twisting Deligne cohomology, during a talk by H.S., which encouraged the authors to revisit and carry out this project. D.G. would like to thank the Mathematics Department at the University of Pittsburgh for hospitality during the final writing of this paper. The authors thank Richard Hain for bringing to their attention related works in algebraic geometry and the referee for useful remarks.

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Authors and Affiliations

  1. 1.Division of Science and MathematicsNew York University NYUADSaadiyat IslandUAE

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