On the structure of residue currents and functionals orthogonal to ideals in the space of holomorphic functions

This article investigates residues associated with holomorphic mappings $f=(f_1,\dots,f_p)\colon X\to\mathbb C^p$ defined on a complex space $X$. By means of a new definition of principal value of a residue, it sharpens results of Coleff, Herrera, and Dolbeault concerning the structure of residues. It establishes a connection between residues and functionals in $\mathcal O'(X)$ orthogonal to the ideal $\langle f_1,\dots,f_p\rangle\subset\mathcal O(X)$. Using these results on residues and functionals, a formula is derived for the exponential representation for elements of invariant subspaces and for the solution of homogeneous convolution equations.