Anti-self dual conformal structures on 4-manifolds

Abstract

The Weyl curvature tensor is the conformally invariant part of the Riemann curvature tensor. On an oriented 4-dimensional manifold, the Weyl tensor is a direct sum of two pieces, W₊ and W., its self and anti-self dual parts. Here is a theorem: for a compact, oriented 4-manifold, M, the connect sum of M with a large enough number of (orientation reversed) CP²’s has a metric with W₊ = 0. by the way, Penrose’s twistor space for such a metric is a complex 3-fold. And so, there are a zoo of bizarre 3-folds.