Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 008, 43 pages      arXiv:2102.10853      https://doi.org/10.3842/SIGMA.2026.008

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

Florian Beck a, Indranil Biswas b, Sebastian Heller c and Markus Röser a
a) Fachbereich Mathematik, Universität Hamburg, 20146 Hamburg, Germany
b) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
c) Beijing Institute of Mathematical Sciences and Applications, No. 544, Hefangkou Village Huaibei Town, Beijing 101408, P.R. China

Received March 25, 2025, in final form January 13, 2026; Published online February 02, 2026

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.

Key words: Deligne-Hitchin twistor space; self-duality equation; connection; circle action; hyperholomorphic bundle.

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