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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 g2, and let MDH be the rank one Deligne-Hitchin moduli space associated to X. It is known that MDH is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on X. We investigate the group Aut(MDH) of all holomorphic automorphisms of MDH. The connected component of Aut(MDH) containing the identity automorphism is computed. There is a natural element of H2(MDH,Z). We also compute the subgroup of Aut(MDH) that fixes this second cohomology class. Since MDH admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that MDH is Moishezon.

Key words: Hodge moduli space; Deligne-Hitchin moduli space; λ-connections; Moishezon twistor space.

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