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Selfdual 4-manifolds, projective surfaces, and the Dunajski-West construction.(English)Zbl 1288.53075

Summary: 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 hyper-Kähler 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.

MSC:

53C80 Applications of global differential geometry to the sciences
32L25 Twistor theory, double fibrations (complex-analytic aspects)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C28 Twistor methods in differential geometry
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism

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