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Weakly normal complex spaces. (English) Zbl 0612.32010
Contributi de Centro Linceo Interdisciplinare di Scienze Matematiche e loro Applicazioni, No. 55. Roma: Accademia Nazionale dei Lincei. 59 p. (1981).
On a complex analytic space (X,$${\mathcal O}_ X)$$ there are, besides $${\mathcal O}_ X$$, two natural sheaves of functions: the sheaf $${\mathcal O}^ c_ X$$ of c-holomorphic and the sheaf $$\hat {\mathcal O}_ X$$ of weakly holomorphic functions, $${\mathcal O}_ X\subset {\mathcal O}^ c_ X\subset \hat {\mathcal O}_ X$$. The space X is said to be weakly normal (w.n.) if $${\mathcal O}_ X={\mathcal O}^ c_ X$$ (normal if $${\mathcal O}_ X=\hat {\mathcal O}_ X$$; so a normal space is also weakly normal). The paper gives a very good survey of the theory of w.n. spaces as developed until 1981, together with some new results, in particular those on locally optimal w.n. spaces and on singularities, the authors being interested in weak normality from the point of view of singularity theory. The two basic techniques used are the Hartogs type extension theorems for coherent sheaves and the Mayer-Vietoris criterion for weak normality of unions.
Section 1 gives a brief introduction to the theory, Section 2 contains examples of weakly normal spaces, including Whitney umbrellas, whose weak normality is proved directly. In Section 3 a theorem of Oka for w.n. spaces is proved. Sections 4 and 5 deal with Hartogs theorems. The class of locally optimal spaces is introduced in Section 6, and Section 7 deals with the weak normality of ordinary singularities.

##### MSC:
 32C20 Normal analytic spaces 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32Sxx Complex singularities