×

Andreotti-Grauert theory by integral formulas. (English) Zbl 0654.32001

Mathematical Research 43. Berlin (GDR): Akademie-Verlag. 270 p. (1988).
The Andreotti-Grauert theory is the theory of the cohomology of q-convex and q-concave manifolds. This monograph develops this theory from the viewpoint of explicit integral representation for solution of the Cauchy- Riemann equation for differential forms on strictly q-convex and q- concave domains. This method give us new proofs of remarkable results with uniform estimates. This results are: uniform approximation and uniform interpolation for \({\bar \partial}\)-cohomology classes on strictly q-convex domains, solution of the Levi problem for \({\bar \partial}\)-cohomology with uniform estimate, Andreotti-Vesentini separation theorem of the Dolbeault cohomology of order q on q-concave manifolds, the Rossi theorem on attaching complex manifolds along a strictly pseudoconcave boundary of real dimension \(\geq 5.\)
Brief contents. Chapter 1. Integral formulas and first applications. Chapter 2. q-convex and q-concave manifolds. Chapter 3. The Cauchy- Riemann equations on q-convex manifolds. Chapter 4. The Cauchy-Riemann equations on q-concave manifolds. Chapter 5. Some applications.
Reviewer: S.Krendelev

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32F10 \(q\)-convexity, \(q\)-concavity
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
PDFBibTeX XMLCite