Summary: Let $X$ and $X'$ be compact Riemann surfaces of genus at least three. Let $G$ and $G'$ be nontrivial connected semisimple linear algebraic groups over $\CC$. If some components $\MDH^d(X,G)$ and $\MDH^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then $X'$ is isomorphic to $X$ or to the conjugate Riemann surface $\Xbar$.

14D20, 14C34

principal bundle, Deligne--Hitchin moduli space, Higgs bundle, vector field