On Saito's normal crossing condition

Journal of Singularities
volume 14 (2016), 124-147

Received: 10 January 2014. Received in revised form: 14 September 2016.

DOI: 10.5427/jsing.2016.14h

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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.

Keywords:

logarithmic differential form, regular differential form, normal crossing divisor, residue, duality

2010 Mathematical Subject Classification:

14H20 (Primary) 14M07, 32A27 (Secondary)

Author(s) information:

Mathias Schulze
Department of Mathematics
University of Kaiserslautern
67663 Kaiserslautern
Germany
email: mschulze@mathematik.uni-kl.de