Weakly holomorphic functions on complete intersections, and their holomorphic extension

The properties of weakly holomorphic functions on analytic sets which are complete intersections are investigated: universal denominators are determined for a system of equations $f=0$ defining the analytic set $A$; a (residual) current $hR_f$ is constructed such that it is $\overline\partial$-closed if and only if the weakly holomorphic function $h$ can be locally extended from $A$; and integral representations for weakly holomorphic functions are given. These results are applied to the problem of lowering the order of poles of rational differential 2-forms in $\mathbf C^2$.
Bibliography: 20 titles.