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

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