Torelli theorem for the Deligne--Hitchin moduli space, II

TL;DR

This paper proves a Torelli-type theorem for Deligne--Hitchin moduli spaces, showing that their biholomorphic structure determines the underlying Riemann surface up to conjugation, extending classical Torelli results.

Contribution

It establishes a Torelli theorem for the Deligne--Hitchin moduli space, linking its biholomorphic structure to the isomorphism class of the underlying Riemann surface.

Findings

Biholomorphic Deligne--Hitchin spaces imply isomorphic or conjugate Riemann surfaces.

The result extends classical Torelli theorems to a new geometric setting.

The theorem applies to moduli spaces associated with nontrivial semisimple groups.

Abstract

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 C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or to the conjugate Riemann surface $\bar{X}$.