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Pseudoconvexity, the Levi problem and vanishing theorems. (English) Zbl 0811.32011
Grauert, H. (ed.) et al., Several complex variables VII. Sheaf- theoretical methods in complex analysis. Berlin: Springer-Verlag. Encycl. Math. Sci. 74, 221-257 (1994).
The author reviews results concerning the topic mentioned in the title. The contents of the paper is as follows:
§1. Plurisubharmonic functions and pseudoconvexity (1. The notion plurisubharmonic functions. 2. Properties of plurisubharmonic functions. 3. Pseudoconvex domains).
§2. Convex spaces (1. Remmert reduction. 2. The Levi problem for 1- convex spaces. 3. Maximal compact analytic sets. 4. Positive sheaves and the normal bundle. 5. The cohomology of 1-convex spaces).
§3. The Levi problem (1. The classical Levi problem. 2. Counterexamples. 3. Characterizing Stein spaces. 4. The local Stein problem).
§4. Positive sheaves and vanishing theorems (1. The projective bundle. 2. The vanishing theorem for positive sheaves. 3. The embedding theorem. 4. Characterization of positivity by cohomology vanishing. 5. Functorial properties of positive sheaves. 6. Differential-geometric positivity notions. 7. Hodge metrics. 8. Relative positive sheaves).
§5. More vanishing theorems (1. Demailly’s vanishing theorem. 2. The notion of \(k\)-ampleness. 3. Grauert-Riemenschneider vanishing theorem and direct images of dualizing sheaves.).
For the entire collection see [Zbl 0793.00010].

32T99 Pseudoconvex domains
32L20 Vanishing theorems
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32E05 Holomorphically convex complex spaces, reduction theory