Torelli theorem for the Deligne–Hitchin moduli space. II

Abstract

Let $X$ and $X^{\prime }$ be compact Riemann surfaces of genus at least three. Let $G$ and $G^{\prime }$ be nontrivial connected semisimple linear algebraic groups over $\mathbb{C}$. If some components ${\mathcal{M}}_{\mathrm{DH}}^d(X,G)$ and ${\mathcal{M}}_{\mathrm{DH}}^{d^{\prime }}(X^{\prime },G^{\prime })$ of the associated Deligne–Hitchin moduli spaces are biholomorphic, then $X^{\prime }$ is isomorphic to $X$ or to the conjugate Riemann surface $\overline{X}$.