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

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.

for Progress in Twistor Theory, SIGMA

References in corpus (5)

• Anti-self-dual four-manifolds with a parallel real spinor
• Integrable Background Geometries
• Anti-self-dual conformal structures with null Killing vectors from projective structures
• Selfdual Einstein metrics and conformal submersions
• Self-dual Zollfrei conformal structures with alpha-surface foliation

Cited by in corpus (10)

• Integrable Background Geometries
• Metrisability of three-dimensional projective structures
• A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence
• Self-dual Zollfrei conformal structures with alpha-surface foliation
• Geometric Theory of Weyl Structures
• Singular Self-dual Zollfrei Metrics and Twistor Correspondence
• Minimal Lagrangian connections on compact surfaces
• Metrisability of projective surfaces and pseudo-holomorphic curves
• The Cayley cubic and differential equations
• Einstein Metrics, Projective Structures and the $SU(\infty )$ Toda Equation