Let$Z$be the zero set of a holomorphic section$f$of a Hermitian vector bundle. It is proved that the current of integration over the irreducible components of$Z$of top degree, counted with multiplicities, is a product of a residue factor$R$^{$f$}and a “Jacobian factor”. There is also a relation to the Monge-Ampère expressions ($dd$^{$c$}log|$f$|)^{$k$}, which we define for all positive powers$k$.

We study to what extent rearrangements preserve the integrability properties of higher order derivatives. It is well known that the second order derivatives of the rearrangement of a smooth function are not necessarily in$L$^{1}. We obtain a substitute for this fact. This is done by showing that the total curvature for the graph of the rearrangement of a function is bounded by the total curvature for the graph of the function itself.

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