Simon Salamon

It is well known that an oriented conformal structure in 2 real dimensions is the same thing as a complex structure. In higher (even) dimensions one can attempt to define different complex structures on the same Riemannian manifold, and their parametrization leads one to « twistor space ».

I shall discuss the simplest case of this theory by describing complex structures on 4-dimensional Euclidean space, and explaining why they are all constant (a « Liouville theorem »). Generalizations of this problem are tackled by passing to complex projective 3-space with a certain real structure. In this context, I shall discuss quadric and cubic surfaces and their discriminant loci.