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Grauert's line bundle convexity; Riemann domain; holomorphic reduction
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 Grauert 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 $\Bbb {P}^n$, where $n$ is the complex dimension of $X$ and $L$ is the pull-back of ${\mathcal O}(1)$.
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