Geometry of the Space of Sections of Twistor Spaces with Circle Action

TL;DR

This paper explores the holomorphic symplectic structure of the space of sections of twistor spaces with circle symmetry, linking Hitchin's meromorphic connection to hyperholomorphic line bundles and analyzing moment maps, especially in Deligne-Hitchin spaces.

Contribution

It provides a new interpretation of Hitchin's meromorphic connection via the Atiyah-Ward transform and studies the critical points of the associated moment map in the context of twistor spaces.

Findings

Residue of the meromorphic connection acts as a moment map.

Critical points of the moment map are characterized.

Application to Deligne-Hitchin moduli spaces.

Abstract

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space carries a line bundle with meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah-Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and furthermore the critical points of this moment map are studied. Particular emphasis is given to the example of Deligne-Hitchin moduli spaces.