A. Elhaouzi, GROTHENDIECK-DOLBEAULT LEMMA FOR COMPLEX ANALYTIC SETS, Comptes rendus de l'Academie des sciences. Serie 1, Mathematique, 323(2), 1996, pp. 179-184
For X, art analytic subset of dimension n in C-N, and r is an element
of [1, n] a positive integer, we show that if phi is a closed C-infini
ty form of type (r, 0) on X, then there exists a holomorphic r-form on
X such that psi = phi on X - Sing(X).