### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 072, 19 pages      arXiv:1704.04924      https://doi.org/10.3842/SIGMA.2017.072

### On the Automorphisms of a Rank One Deligne-Hitchin Moduli Space

Indranil Biswas a and Sebastian Heller b
a) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
b) Institut für Differentialgeometrie, Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

Received May 13, 2017, in final form September 01, 2017; Published online September 06, 2017

Abstract
Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element of $H^2({\mathcal M}_{\rm DH}, {\mathbb Z})$. We also compute the subgroup of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ that fixes this second cohomology class. Since ${\mathcal M}_{\rm DH}$ admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that ${\mathcal M}_{\rm DH}$ is Moishezon.

Key words: Hodge moduli space; Deligne-Hitchin moduli space; $\lambda$-connections; Moishezon twistor space.

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References

1. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
2. Baraglia D., Classification of the automorphism and isometry groups of Higgs bundle moduli spaces, Proc. Lond. Math. Soc. 112 (2016), 827-854, arXiv:1411.2228.
3. Baraglia D., Biswas I., Schaposnik L.P., Automorphisms of $\mathbb{C}^*$ moduli spaces associated to a Riemann surface, SIGMA 12 (2016), 007, 14 pages, arXiv:1508.06587.
4. Corlette K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382.
5. Donaldson S.K., Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131.
6. Goldman W.M., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200-225.
7. Goldman W.M., Xia E.Z., Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces, Mem. Amer. Math. Soc. 193 (2008), viii+69 pages, math.DG/0402429.
8. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
9. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
10. Simpson C., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
11. Simpson C., The Hodge filtration on nonabelian cohomology, in Algebraic Geometry - Santa Cruz 1995, Proc. Sympos. Pure Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1997, 217-281, alg-geom/9604005.
12. Simpson C., A weight two phenomenon for the moduli of rank one local systems on open varieties, in From Hodge Theory to Integrability and TQFT tt*-Geometry, Proc. Sympos. Pure Math., Vol. 78, Amer. Math. Soc., Providence, RI, 2008, 175-214, arXiv:0710.2800.
13. Verbitsky M., Holography principle for twistor spaces, Pure Appl. Math. Q. 10 (2014), 325-354, arXiv:1211.5765.
14. Weil A., Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1957 (1957), 33-53.