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

SIGMA 10 (2014), 035, 18 pages      math.DG/0606754      https://doi.org/10.3842/SIGMA.2014.035
Contribution to the Special Issue on Progress in Twistor Theory

### Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction

David M.J. Calderbank
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received January 21, 2014, in final form March 18, 2014; Published online March 28, 2014

Abstract
I present a construction of real or complex selfdual conformal 4-manifolds (of signature (2,2) in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex 2-manifold. The 4-manifolds obtained are characterized by the existence of a foliation by selfdual null surfaces of a special kind. The classification by Dunajski and West of selfdual conformal 4-manifolds with a null conformal vector field is the special case in which the gauge group reduces to the group of diffeomorphisms commuting with a vector field, and I analyse the presence of compatible scalar-flat Kähler, hypercomplex and hyperkähler structures from a gauge-theoretic point of view. In an appendix, I discuss the twistor theory of projective surfaces, which is used in the body of the paper, but is also of independent interest.

Key words: selfduality; twistor theory; integrable systems; projective geometry.

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