Non Completely Solvable Systems of Complex First Order PDE's

Abstract

We revisit the lack of local solvability for homogeneous vector fields with smooth complex valued coefficients, in the spirit of Nirenberg's three dimensional example. First we provide a short expository proof, in the case of CR dimension one, with arbitrary CR codimension. Next we pass to Lorenzian structures with any CR codimension $\ge 1$ and CR dimension $\ge 2$. Several different approaches are presented. Finally we discuss the connection with the absence of the Poincare lemma and the failure of local CR embeddability, and present a global example.