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Grauert’s line bundle convexity, reduction and Riemann domains. (English) Zbl 1389.32021

Summary: We consider a convexity notion for complex spaces \(X\) with respect to a holomorphic line bundle \(L\) over \(X\). This definition has been introduced by H. Grauert [Math. Z. 81, 377–391 (1963; Zbl 0151.09702)] and, when \(L\) is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if \(H^0(X,L)\) separates each point of \(X\), then \(X\) can be realized as a Riemann domain over the complex projective space \(\mathbb{P}^n\), where \(n\) is the complex dimension of \(X\) and \(L\) is the pull-back of \(\mathcal{O}(1)\).

MSC:

32F17 Other notions of convexity in relation to several complex variables
32E05 Holomorphically convex complex spaces, reduction theory
32E99 Holomorphic convexity

Citations:

Zbl 0151.09702
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References:

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