Diederich, Klas; Fornaess, John Erik Smoothing q-convex functions and vanishing theorems. (English) Zbl 0586.32022 Invent. Math. 82, 291-305 (1985). It is well known that the supremum of a finite number of p.s.h. functions (1-convex functions) is again p.s.h. but that is not true in general for q-convex functions with \(q>1\). This paper shows nevertheless that if a complex manifold M has a continuous exhaustion function which is, outside a compact set K, locally the supremum of a finite number of \(C^{\infty}\) q-convex functions, then the manifold M is \(\hat q-\)convex (in the sense of Andreotti-Grauert) with \(\hat q=n-[\frac{n}{q}]+1\) where \(n=\dim_{{\mathbb{C}}}M.\) Moreover, if \(K=\emptyset\) M is \(\hat q-\)complete. This leads to interesting new vanishing theorems. (For the singular case see, by the same authors, Math. Ann. 273, 665-671 (1986; see the following review)). Reviewer: D.Barlet Cited in 1 ReviewCited in 28 Documents MSC: 32F10 \(q\)-convexity, \(q\)-concavity 32L20 Vanishing theorems 32Q99 Complex manifolds Keywords:q-convexity with corners; vanishing theorems Citations:Zbl 0586.32023 PDFBibTeX XMLCite \textit{K. Diederich} and \textit{J. E. Fornaess}, Invent. Math. 82, 291--305 (1985; Zbl 0586.32022) Full Text: DOI EuDML References: [1] Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr.90, 193-259 (1962) · Zbl 0106.05501 [2] Barth, W.: Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projektiven Raum. Math. Ann.187, 150-162 (1970) · Zbl 0189.36704 · doi:10.1007/BF01350179 [3] Barth, W.: Über die analytische CohomologiegruppeH n?1 (? n ?A,F). Invent. Math.9, 135-144 (1970) · Zbl 0186.40403 · doi:10.1007/BF01404553 [4] Barth, W.: Transplanting cohomology classes in complex projective space. Am. J. Math.92, 951-967 (1970) · Zbl 0206.50001 · doi:10.2307/2373404 [5] Buchner, M., Fritzsche, K., Sakai, T.: Geometry and cohomology of certain domains in complex projective space. J. Reine Angew. Math.323, 1-52 (1981) · Zbl 0447.32003 [6] Buchner, M., Fritzsche, K.: On the dimension of analytic cohomology. Math. Z.179, 375-385 (1982) · Zbl 0477.32009 · doi:10.1007/BF01215340 [7] Buchner, M., Fritzsche, K.: The cohomological and analytic completeness of ?(?2? n )?G 2n . Math. Ann.261, 327-338 (1982) · Zbl 0489.32005 · doi:10.1007/BF01455454 [8] Faltings, G.: Über lokale Kohomologiegruppen hoher Ordnung. J. Reine Angew. Math.313, 43-51 (1980) · Zbl 0411.13010 · doi:10.1515/crll.1980.313.43 [9] Fritzsche, K.:q-konvexe Restmengen in kompakten komplexen Mannigfaltigkeiten. Math. Ann.229, 251-273 (1976) · Zbl 0327.32007 · doi:10.1007/BF01596392 [10] Fritzsche, K.: Pseudoconvexity properties of complements of analytic subvarieties. Math. Ann.230, 107-122 (1977) · Zbl 0353.32022 · doi:10.1007/BF01370658 [11] Grauert, H.: Kantenkohomologie. Comput. Math.44, 79-101 (1981) · Zbl 0512.32011 [12] Peternell, M.: Ein Lefschetz-Satz für Schnitte in projektiv-algebraischen Mannigfaltigkeiten. Math. Ann.264, 361-388 (1981) · Zbl 0514.55017 · doi:10.1007/BF01459131 [13] Peternell, M.: Homotopie in homogenen komplexen Mannigfaltigkeiten. Math. Z.188, 271-278 (1985) · Zbl 0552.57018 · doi:10.1007/BF01304214 [14] Peternell, M.: Continuousq-convex exhaustion functions. (Preprint 1985) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.