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

Abstract

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

References in corpus (1)

• Automorphisms of ${\mathbb{C}}^{\ast }$ Moduli Spaces Associated to a Riemann Surface

Cited by in corpus (3)

• Real holomorphic sections of the Deligne-Hitchin twistor space
• Geometry of the Space of Sections of Twistor Spaces with Circle Action
• Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties