Vol. 52, No. 2, 1974

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A characterization of normal analytic spaces by the homological codimension of the structure sheaf

Andrew Guy Markoe

Vol. 52 (1974), No. 2, 485–489

In 1951 K. Oka proved that if X is a hypersurface in Cn whose singular set Σ(X) has codimension at least 2 in X, then X is a normal analytic space. This result was subsequently generalized by S. Abhyankar and (independently) W. Thimm to the case of a complete intersection.

The main result of the present work is the following criterion for normality: If dim[Σ(X) ∩{x X : codhxPX k + 2}] k for all integers k 1, then X is a normal analytic space. This is the best possible criterion following the lines of Oka, Abhyankar, and Thimm, and is, in fact, a characterization; the converse is true. The criterion implies that whenever pX has Cohen-Macaulay stalks and Σ(X) has codimension at least 2, then X is normal. Finally, the techniques used in proving the criterion are used to obtain a vanishing theorem for the first cohomology group of the complement of a subvariety A of suitably high codimension in a Stein manifold, with coefficients in the ideal sheaf of a normal subvariety containing A.

Mathematical Subject Classification 2000
Primary: 32C20
Received: 3 July 1973
Revised: 8 February 1974
Published: 1 June 1974
Andrew Guy Markoe