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Characterizing normal crossing hypersurfaces. (English) Zbl 1333.32034
The purpose of this paper is to give an algebraic characterization of normal crossing divisors on complex manifolds. It is known that a normal crossing divisor is a free divisor. The author proves that a divisor on a complex manifold has normal crossings at a point if and only if it is free with radical Jacobian ideal at that point and its normalization is smooth. Since there is a characterization of free divisors using their Jacobian ideals [A. G. Aleksandrov, Math. USSR, Sb. 65, No. 2, 561–574 (1990; Zbl 0684.32010); translation from Mat. Sb., Nov. Ser. 137(179), No. 4(12), 554–567 (1988)], one obtains a purely algebraic characterization of normal crossing divisors.
From K. Saito’s theory [J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 265–291 (1980; Zbl 0496.32007)], the characterization can be given in terms of logarithmic differential forms and vector fields, and also from [M. Granger and M. Schulze, Compos. Math. 150, No. 9, 1607–1622 (2014; Zbl 1314.32043)] it can be given in terms of the logarithmic residue.

MSC:
32S25 Complex surface and hypersurface singularities
32S10 Invariants of analytic local rings
32A27 Residues for several complex variables
14B05 Singularities in algebraic geometry
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