Real holomorphic sections of the Deligne-Hitchin twistor space

TL;DR

This paper investigates holomorphic sections of the Deligne-Hitchin moduli space invariant under anti-holomorphic involutions, linking them to harmonic maps and addressing a question posed by Simpson.

Contribution

It establishes the relationship between invariant holomorphic sections and harmonic maps, and answers a question posed by Simpson regarding these sections.

Findings

Invariant holomorphic sections correspond to certain harmonic maps.

The paper provides a classification of these sections under involutions.

It resolves a specific question raised by Simpson about such sections.

Abstract

We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered.