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 {\it CR} dimension one, with arbitrary {\it CR} codimension. Next we pass to Lorenzian structures with any {\it CR} codimension $\geq 1$ and {\it CR} dimension $\geq 2$. Several different approaches are presented. Finally we discuss the connection with the absence of the Poincare lemma and the failure of local {\it CR} embeddability, and present a global example.