On the Stein coverings for 1-convex spaces.
This paper studies compactifications and modifications of complex varieties in the framework of \(1\)-convexity and Moishezon spaces. The theorems are as follows. Theorem 1.1. Let \(X\) be a \(1\)-convex space with exceptional set \(S\), \(L\to X\) a holomorphic line bundle. Then the following are equivalent. (a) \(L\) is weakly positive (i.e., a metric condition holds). (b) \(L\) is cohomologically positive (i.e., twists of any coherent analytic sheaf over \(X\) by high enough powers of \(L\) are acyclic). (c) \(L\) is ample. Theorem 2.1. Let \(X,S\) be as in Theorem\,1.1, and suppose that \(X\subset{\mathbb C}^k\times{\mathbb P}^\nu\) embeds in mixed linear space. Then \(X\) can be covered by \(q=\dim(S)+1\) of its Stein open subsets. The proof of Theorem 2.1 is by induction on \(\dim(S)\) and goes roughly as follows. The case \(\dim(S)=0\) is essentially tautological, and the claim of Theorem 2.1 holds if \(X\) is a hyperplane \(Y\) in \({\mathbb C}^k\times{\mathbb P}^\nu\), and in general one can look at a hyperplane section \(X\cap Y\) of \(X\) and thicken up a Stein open covering \(\{U_j\}\) in \(X\cap Y\) of \(X\cap Y\) by a theorem of Siu on Stein neighborhoods to an open Stein covering \(\{V_j\}\) of \(X\cap Y\) in \(X\), and add \(X\setminus(X\cap Y)\) to the covering, which also turns out to be Stein open in \(X\). Theorem 2.2. Let \(X\) be a complex algebraic space with \(H^1(X,F)\) finite dimensional for each coherent algebraic sheaf \(F\to X\). Let \(A\to X\) be a coherent algebraic sheaf, and \(A'\) the corresponding coherent analytic sheaf over the complex analytic space \(X'\) underlying \(X\). Then (a) \(X'\) is a \(1\)-convex space, and (b) the natural map of cohomology groups \(H^i(X,A)\to H^i(X',A')\) is bijective for \(i\geq1\). \S\,3 talks about affine covers of projectivizable \(1\)-convex algebraic spaces. The latter are complex spaces algebraically isomorphic to a space of the form \(M\setminus D\), where \(M\) is a projective algebraic space and \(D\subset M\) is a normally ample divisor. Theorem 3.2. An irreducible algebraic \(1\)-convex space \(X\) with exceptional set \(S\) is projectivizable if and only if \(X\) can be covered by \(q=\dim(S)+1\) of its Zariski open affine subsets. The paper concludes with some questions and with a discussion of the various strict or possibly not strict containment relations among the families of projectivizable, algebraic, quasi-algebraic, quasi-projective, and compactifiable \(1\)-convex spaces. The write-up and typing of the paper could have been more careful.
scientific article; zbMATH DE number 1941864