Removable Singularities for Positive Currents

Suppose A is a closed subset of an open set Q contained in Cn. The main result (Theorem 6) of this paper is about extending currents defined on -A across A. More precisely, suppose u is a current defined on Q A which is d-closed and of complex dimension m. If the 2n -1 dimensional Hausdorff imeasure of A vanishes then the current u can be extended across A. If V is a pure m-dimensional subvariety of Q-A then integration over the regular points of V defines an m-dimensional, positive, d-closed current on Q -A. Theorem 6 can be used to prove Shiffman's Theorem 9 which says that if V is a pure m-dimensional subvariety of Q A and the 2m 1 dimensional IHausdorff measure of A vanishes then V is a pure m-dimensional subvariety of Q. Of course, if A itself is a subvariety of Q2 of complex dimension m-1 or less, then the 2m-1 dimensional Hausdorff measure of A vanishes. That is, Shiffman's theorem includes the well known theorem of Remmert and Stein 8 as a special case. A KDhler form oln? Q-A provides another example of a positive, dclosed current (of complex dimension n 1 or type (1, 1)). Consequently, Theorem 6 can be used to give a unified proof of a result (Theorem 8) on extending holomorphic maps of Q -A into Kahler manifolds across A; which was first proved by Gritffihs 2 for codimension A ? 3 and then by Shiffman 10 for codimension A = 2. As another application of Theorem 6 we obtain the result (Theorem 9) that positive line bundles defined outside a set A of 2n -3 dimensional Hausdorff measure zero extend across A (the extension is also positive). Shiffman 10 proved this results for a subvariety A of dimension n -2 or less. It is my pleasure to acknowledge several helpful conversations with John Polking.