A twistor transform and normal forms for Cauchy Riemann structures

Abstract We use Hitchin’s twistor transform for two-dimensional projective structures to obtain normal coordinates in a pseudoconcave neighbourhood of an O ⁢ ( 1 ) \mathcal{O}(1) rational curve; in the construction, we present every such neighbourhood as Q D / F Q_{\mathbb{D}}/\mathcal{F} for some holomorphic foliation ℱ, where Q D Q_{\mathbb{D}} is an open neighbourhood in the standard quadric Q ⊂ P 2 × P 2 Q\subset\mathbb{P}^{2}\times\mathbb{P}^{2} . As a consequence of the normal coordinates, we obtain a new normal form for Cauchy Riemann structures on the three-sphere that are isotopic to the standard one. We end the paper with explicit calculations for the cases arising from deformations of the normal isolated singularities X ⁢ Y = Z n XY=Z^{n} .