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Transactions of the American Mathematical Society

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Monodromy of rank 2 twisted Hitchin systems and real character varieties


Authors: David Baraglia and Laura P. Schaposnik
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14H60, 53C07; Secondary 14H70
DOI: https://doi.org/10.1090/tran/7144
Published electronically: February 28, 2018
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Abstract: We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $ L$-twisted $ G$-Higgs bundles for the groups $ G = GL(2,\mathbb{C})$, $ SL(2,\mathbb{C})$, and $ PSL(2,\mathbb{C})$. We also determine the Tate-Shafarevich class of the abelian torsor defined by the regular locus, which obstructs the existence of a section of the moduli space of $ L$-twisted Higgs bundles of rank $ 2$ and degree $ \deg (L)+1$. By counting orbits of the monodromy action with $ \mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups $ G = GL(2,\mathbb{R})$, $ SL(2,\mathbb{R})$, $ PGL(2,\mathbb{R})$, $ PSL(2,\mathbb{R})$, as well as the number of components of the $ Sp(4,\mathbb{R})$ and $ SO_0(2,3)$-character varieties with maximal Toledo invariant. We also use our results for $ GL(2,\mathbb{R})$ to compute the monodromy of the $ SO(2,2)$ Hitchin map and determine the components of the $ SO(2,2)$ character variety.


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Additional Information

David Baraglia
Affiliation: School of Mathematical Sciences, The University of Adelaide, Adelaide SA 5005, Australia
Email: david.baraglia@adelaide.edu.au

Laura P. Schaposnik
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607; and Department of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany
Email: schapos@uic.edu

DOI: https://doi.org/10.1090/tran/7144
Received by editor(s): January 21, 2016
Received by editor(s) in revised form: November 28, 2016
Published electronically: February 28, 2018
Additional Notes: The work of the first author was supported by the Australian Research Council Discovery Project DP110103745.
The work of the second author was supported by the Simons Foundation through an AMS-Simons Travel Grant.
Article copyright: © Copyright 2018 American Mathematical Society

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