Criteria for holomorphic completeness.~II

V. D. Golovin

Izvestiya. Mathematics , Tome 59 (1995) no. 4, pp. 671-676

Résumé

It is proved that a complex space $X$ of finite dimension $d$ is holomorphically complete if and only if the following conditions hold: 1) for an arbitrary point $x_0\in X$ there exists analysis sets $M_n\subset\dots\subset M_1\subset M_0=X$ and holomorphic function $f_i\in\Gamma(M_{i-1};\mathscr O_{M_{i-1}})$, $i=1,\dots,n$, such that $M_i=\{x\in M_{i-1}:f_i(x)=0\}$, and $\mathscr O_{M_i}=\mathscr O_{M_{i-1}}/f_i\mathscr O_{M_{i-1}}\mid M_i$ for each $i=1,\dots,n$, and $x_0$ is an isolated point in $M_n$; 2) $H^k(X;\mathscr O_X)=0$, for $k=1,\dots,d-1$.