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

SIGMA 17 (2021), 102, 11 pages      arXiv:2108.01739      https://doi.org/10.3842/SIGMA.2021.102
Contribution to the Special Issue on Twistors from Geometry to Physics in honor of Roger Penrose

Twistors, Self-Duality, and Spin$^c$ Structures

Claude LeBrun
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651 USA

Received August 02, 2021, in final form November 15, 2021; Published online November 19, 2021

Abstract
The fact that every compact oriented 4-manifold admits spin$^c$ structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spin$^c$ structures in any dimension.

Key words: 4-manifold; spinc structure; twistor space; self-dual 2-form.

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