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Non completely solvable systems of complex first order PDE’s. (English) Zbl 1275.35074

Summary: 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 \(\geq 1\) and \(CR\) 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 \(CR\) embeddability, and present a global example.

MSC:

35F05 Linear first-order PDEs
32V05 CR structures, CR operators, and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
17B20 Simple, semisimple, reductive (super)algebras
57T20 Homotopy groups of topological groups and homogeneous spaces
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References:

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