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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.
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