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Twisted smooth Deligne cohomology. (English) Zbl 1398.58002

Twisted Deligne cohomology is the prototype for other twisted differential spectra, and its existence follows from general constructions. For example, a sheaf-theoretic definition of smooth Deligne cohomology was given in [U. Bunke, “Differential cohomology”, Preprint, arXiv:1208.3961], and a bordism model for the differential extension of ordinary integral cohomology was presented in [U. Bunke et al., Ann. Math. Blaise Pascal 17, No. 1, 1–16 (2010; Zbl 1200.55007)]. Since Deligne cohomology has not been explicitly twisted like other differential cohomology theories, in the paper under review the authors aim to twist Deligne cohomology, by using degree one twists of integral cohomology and de Rham cohomology. The main tools are the homotopy sheaves, simplicial presheaves and higher stacks.
In Section 2, the authors deal with twists of integral cohomology at the level of the Eilenberg-Mac Lane spectrum \(H\mathbb{Z}\), and in Section 3 they recall the properties of the Deligne cohomology. Then, in Section 4, twisted integral cohomology and 1-form twisted de Rham cohomology yield compatible twistings of Deligne cohomology. The authors prove that pulling back the universal bundle over a map which classifies a twist, one obtains a bundle \(\mathcal{H}^q \rightarrow M\) over \(M\). The \(\omega\)-twisted Deligne cohomology of degree \(q\) of \(M\) is given by the connected components \(\pi_0\Gamma(M,\mathcal{H}^q)\). In the paper under review the authors use the category of sheaves of chain complexes, but for some results they deal also with twisted differential cohomology within smooth sheaves of spectra.
The properties of basic twisted Deligne cohomology are presented in Section 5, and several examples for the constructions and techniques developed in the previous sections are given in the final part of the paper.

MSC:

58A12 de Rham theory in global analysis
14F40 de Rham cohomology and algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

Citations:

Zbl 1200.55007
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References:

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