Compact sets with vanishing cohomology in Stein spaces and domains of holomorphy

Vâjâitu, Viorel

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 3, pp. 665-685.

Let $X$ be a Stein space. We study compact subsets $K$ of $X$ that are structurally acyclic, i.e., $H^i(K,{𝒪}_X)=0$, for all $i\ge 1$. We show i) that such compact sets are natural in the sense that the canonical map from $K$ into $\tilde{K}$, the spectrum of the complex algebra $\Gamma (K,{𝒪}_X)$, is bijective, and ii) that the set of interior points of $K$ is a domain of holomorphy in $X$. Motivated by this we give an extensive account of examples of domains of holomorphy in non-normal Stein spaces and prove several properties, like hereditarity via the normalization map. Finally, a straightforward criterion of non-acyclicity is given in terms of general Hartogs figures.

Classification : 32D05, 32C35, 32E10, 32C15

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