On Saito's normal crossing condition

Abstract: Kyoji Saito defined a residue map from the logarithmic differential 1-forms along a reduced complex analytic hypersurface to the meromorphic functions on the hypersurface. He studied the condition that the image of this map coincides with the weakly holomorphic functions, that is, with the functions on the normalization. With Michel Granger, the author proved that this condition is equivalent to the hypersurface being normal crossing in codimension one. In this article, the condition is given a natural interpretation in terms of regular differential forms beyond the hypersurface case. For reduced equidimensional complex analytic spaces which are free in codimension one, the geometric interpretation of being normal crossing in codimension one is shown to persist.