The Sheaf

Daniel Barlet

Note that erratum exists for this article.

Journal of Singularities
volume 18 (2018), 50-83

Received: 20 August 2017. Accepted: 9 June 2018.

DOI: 10.5427/jsing.2018.18e

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Abstract:

We introduce, in a reduced complex space, a "new coherent sub-sheaf" of the sheaf \omega_X^\bullet which has the "universal pull-back property" for any holomorphic map, and which is, in general, bigger than the usual sheaf of holomorphic differential forms (\Omega_X^\bullet)/torsion. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf (\Omega_X^\bullet)/torsion. This sheaf \alpha_X^\bullet is closely related to the normalized Nash transform. We also show that these q-meromorphic differential forms are locally square-integrable on any q-dimensional cycle in X and that the corresponding functions obtained by integration on an analytic family of q-cycles are locally bounded and continuous on the complement of a closed analytic subset.

Keywords:

Meromorphic, differential forms, singular space, Universal pull-back property, Normalized Nash transform, Integral dependence equation, differential forms

Mathematical Subject Classification:

32C15, 32C30, 32Sxx, 32S45

Author(s) information:

Daniel Barlet
Institut Elie Cartan, Géomètrie
Université de Lorraine
CNRS UMR 7502 and Institut Universitaire de France