Torelli theorem for the Deligne--Hitchin moduli space

Abstract

Fix integers $g\ge 3$ and $r\ge 2$, with $r\ge 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $\MDH(X)$ denote the corresponding $\text{SL}(r,\mathbb{C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $\MDH(X)$ determines (up to an isomorphism) the unordered pair $\{X,\overline{X}\}$, where $\overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$.

14 pages