On generalized Cauchy Riemann equations on manifolds. (English) Zbl 0575.32028

Bureš, Jarolím; Souček, Vladimír

Suppl. Rend. Circ. Mat. Palermo, II. Ser. 6, 31-42 (1984).

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

MSC:

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

Keywords:

generalization of Cauchy-Riemann equations; almost complex structure; Riemann surface; de Rham sequence