On holomorphically complete complex spaces (Q1589403)

The author develops a new approach to the study of holomorphically complete complex spaces, which is based on the homology theory of analytic sheaves [\textit{V.~D.~Golovin}, Homology of Analytic Sheaves and Duality Theorems (Russian), Nauka, Moscow (1986; Zbl 0585.32005)]. The following theorem is proved: Every locally holomorphically trivial analytic sheaf \(F\) over a holomorphically complete space \(X\) is homologically trivial. It follows from this that every coherent analytic sheaf \(F\) over a holomorphically complete complex space \(X\) is homologically trivial. Cartan's fundamental theorems (A) and (B) [\textit{H. Cartan}, Variétés analytiques complexes et cohomologie, Colloque fonctions plusieurs variables, Bruxelles 1953, 41-45 (1953; Zbl 0053.05301)], and the Runge approximation theorem for sections of coherent analytic sheaves are direct consequences of this theorem.