Torelli Theorem for the Deligne–Hitchin Moduli Space

• DOI: 10.1007/s00220-009-0831-3
• Publication Date: 2009-05-05T22:04:14Z
• Authors: Indranil Biswas; Tomás L. Gómez; Norbert Hoffmann; Marina Logares
• Keywords: Isomorphism (crystallography); Compact Riemann surface

Fix integers $g\geq 3$ and $r\geq 2$, with 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 complex analytic space determines (up to an isomorphism) unordered pair $\{X, \overline{X}\}$, where $\overline{X}$ is defined by opposite almost structure on $X$.