On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space

Let X be a compact connected Riemann surface of genus g ≥ 2, and let MDH be the rank one Deligne–Hitchin moduli space associated to X. It is known thatMDH 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 H(MDH,Z). We also compute the subgroup of Aut(MDH) that fixes this second cohomology class. SinceMDH admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that MDH is Moishezon.