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November 2017 A twisted nonabelian Hodge correspondence
Alberto García-Raboso
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J. Differential Geom. 107(3): 455-518 (November 2017). DOI: 10.4310/jdg/1508551223


We prove an extension of the nonabelian Hodge theorem [Sim92] in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beĭlinson and Bernstein [BB93]; on the other side, we endow them with appropriately defined twisted Higgs data.

The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we use homotopy-theoretic methods—chief among them the theory of principal $\infty$-bundles [NSS12a]—to reduce our statement to classical, untwisted Hodge theory [Sim02].


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Alberto García-Raboso. "A twisted nonabelian Hodge correspondence." J. Differential Geom. 107 (3) 455 - 518, November 2017.


Received: 22 April 2013; Published: November 2017
First available in Project Euclid: 21 October 2017

zbMATH: 06846969
MathSciNet: MR3715347
Digital Object Identifier: 10.4310/jdg/1508551223

Primary: 58A14
Secondary: 14C30 , 14D23 , 32J25 , 58A12

Keywords: gerbes , nonabelian Hodge theory , principal $\infty$-bundles

Rights: Copyright © 2017 Lehigh University

Vol.107 • No. 3 • November 2017
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