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On generalized Cauchy Riemann equations on manifolds. (English) Zbl 0575.32028
This is a research announcement, without proofs, of results concerning a generalization of \(\partial\), \({\bar \partial}\) on a Riemann surface to higher dimensional manifolds. The point is that the generalization applies to other than complex manifolds. The concept that replaces ”almost complex structure” is splitting of a vector-valued de Rham sequence.
The authors give several nice applications of their ideas which provide good motivation for the work.
Reviewer: S.G.Krautz

32K99 Generalizations of analytic spaces
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
58J10 Differential complexes
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
14F40 de Rham cohomology and algebraic geometry