# Holomorphically-convex complex space

A complex space $X$ that satisfies the following condition: For each compactum $K \subset X$ the set

$$\left \{ {x \in X } : {| f ( x) | \leq \sup _ { K } | f | ( f \in A) } \right \} ,$$

where $A$ is the algebra of holomorphic functions on $X$, is compact. A space $X$ is holomorphically convex if and only if it admits a proper surjective holomorphic mapping $\phi$ onto some Stein space (a holomorphically-complete space) $\widetilde{X}$ which induces an isomorphism between the algebras of holomorphic functions on these spaces. The mapping $\phi : X \rightarrow \widetilde{X}$( the holomorphic reduction of $X$) is uniquely defined and has connected fibres [1]. For any coherent analytic sheaf $F$ on a holomorphically-convex complex space $X$, the cohomology spaces $H ^ {p} ( X, F )$ and $H _ {c} ^ {p} ( X, F )$, $p \geq 0$, are separable topological vector spaces [2].

A special class of holomorphically-convex complex spaces is formed by the complex spaces $X$ for which the holomorphic reduction mapping is bijective outside some compact analytic set (such a space is obtained from a Stein space by a proper modification which blows-up a finite number of points). A complex space possesses this property if and only if

$$\mathop{\rm dim} H ^ {p} ( X, F ) < \infty ,\ \ p > 0,$$

for any coherent analytic sheaf $F$ on $X$[3]. This class of complex spaces also coincides with the class of strictly $1$- convex complex spaces (cf. Pseudo-convex and pseudo-concave).

#### References

 [1] H. Cartan, "Quotients of complex analytic spaces" , Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960 , Tata Inst. (1960) pp. 1–15 [2] J.P. Ramis, "Théorèmes de séperation et de finitude pour l'homologie et la cohomologie des espaces -convexes-concaves" Ann. Scuola Norm. Sup. Pisa Ser. 3 , 27 (1973) pp. 933–997 [3] R. Narasimhan, "The Levi problem for complex spaces II" Math. Ann. , 146 (1962) pp. 195–216