Torelli theorem for the Deligne--Hitchin moduli space

TL;DR

This paper proves that the Deligne--Hitchin moduli space associated with a Riemann surface uniquely determines the surface and its conjugate, extending Torelli-type results to this complex geometric setting.

Contribution

It establishes a Torelli theorem for the Deligne--Hitchin moduli space, showing it encodes the surface and its conjugate up to isomorphism.

Findings

The Deligne--Hitchin moduli space determines the Riemann surface and its conjugate.

The result applies for genus g ≥ 3 and rank r ≥ 2 (r ≥ 3 if g=3).

Provides a Torelli-type theorem in complex geometry.

Abstract

Fix integers $g\geq 3$ and $r\geq 2$, with $r\geq 3$ 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 $X$.