Characterizing normal crossing hypersurfaces

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a divisor (=hypersurface) has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito’s theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found. Finally, we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.