In this paper the concept of
normality for a complex analytic space X is strengthened to the requirement that
every local holomorphic p-form, for all 0 ≤ p ≤some integer k, defined on the
regular points of X extend across the singular variety. A condition for when this
occurs is given in terms of a notion of independence, in the exterior algebra ΩΔN∗, of
the differentials dF1,⋯,dFr of local generating functions Fi of the ideal of X in some
ambient polydisc ΔN⊂ CN. One result is that for a complete intersection,
“k-independent implies (k − 2)-normal” (precise definitions are given below), which
extends some ideas of Oka, Abhyankar, Thimm, and Markoe on criteria for
normality.
