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

Symmetry Integrability and Geometry-Methods and Applications

article

David M.J. Calderbank¹ 

[1] Department of Mathematical Sciences, University of Bath

关键词: selfduality;    twistor theory;    integrable systems;    projective geometry;   

DOI  :  10.3842/SIGMA.2014.035

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown