Characterizing normal crossing hypersurfaces

TL;DR

This paper provides algebraic characterizations of normal crossing hypersurfaces in complex manifolds, linking free divisors, Jacobian ideals, and smooth normalization, and offers multiple criteria based on logarithmic forms and residues.

Contribution

It introduces new algebraic and differential criteria for identifying normal crossing hypersurfaces, expanding the theoretical framework with recent residue results.

Findings

Normal crossing hypersurfaces are characterized as free divisors with radical Jacobian ideals and smooth normalization.

Multiple equivalent characterizations are provided using logarithmic differential forms, vector fields, and residues.

The results unify algebraic and differential perspectives on hypersurface singularities.

Abstract

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a 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 and and finally another one in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.