×

Differential complexes with nonsmoothable cohomology: obstructions to de Rham’s theorem. (English) Zbl 1134.58008

Summary: Given a locally integrable structure \(T'\) (e.g., a CR structure) on a smooth manifold \(\mathcal M\), there is an associated differential complex of \(C^\infty\) differential \(p,q\)-forms and an associated differential complex of \(p,q\)-currents. The cohomology of a differential complex of \(p,q\)-currents is said to be smoothable if it is isomorphic to the cohomology of the corresponding complex of \(C^\infty\) differential \(p,q\)-forms. By the de Rham theorem, the cohomology of the de Rham complex of currents on a real manifold and the cohomology of the \(\overline\partial\)-complex of \(p,q\)-currents on a complex manifold are both smoothable.
The necessary conditions for the smoothability of differential complexes associated with locally integrable structures are studied. The results and methods are similar to the ones for the vanishing of cohomology in [P. Cordaro and F. Trèves, Proc. Symp. Pure Math. 52, Part 3, 83–91 (1991; Zbl 0744.35007) and P. Cordaro and J. Hounie, Am. J. Math. 112, No. 2, 243–270 (1990; Zbl 0708.58025)].

MSC:

58J10 Differential complexes
35N10 Overdetermined systems of PDEs with variable coefficients
PDFBibTeX XMLCite