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-K\"ahler 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.
Categories: math.AG