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Smoothing q-convex functions and vanishing theorems. (English) Zbl 0586.32022

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

MSC:

32F10 \(q\)-convexity, \(q\)-concavity
32L20 Vanishing theorems
32Q99 Complex manifolds

Citations:

Zbl 0586.32023
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References:

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