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Pulling back singularities of codimension one objects

Authors: Luis Giraldo and Roland K. W. Roeder
Journal: Proc. Amer. Math. Soc. 148 (2020), 1207-1217
MSC (2010): Primary 32SXX, 32S65, 14J17
Published electronically: October 18, 2019
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Abstract: We prove that the preimage of a germ of a singular analytic hypersurface under a germ of a finite holomorphic map $ g: (\mathbb{C}^n,{\bm 0}) \rightarrow (\mathbb{C}^n,{\bm 0})$ is again singular. This provides a generalization of previous results of this nature by Ebenfelt-Rothschild [Comm. Anal.Geom. 15 (2007), no. 2, 491-507], Lebl [ArXiv preprint https://arxiv.orglabs/0812-2498], and Denkowski [Manuscripta Math. 149 (2016), no. 1-2, 83-91]. The same statement is proved for pullbacks of singular codimension one holomorphic foliations.

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Luis Giraldo
Affiliation: Departmento de Álgebra, Geometría y Topología and IMI, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Ciudad Universitaria, Plaza de Ciencias 3, 28040 Madrid, Spain

Roland K. W. Roeder
Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, LD Building, Room 224Q, 402 North Blackford Street, Indianapolis, Indiana 46202-3267

Received by editor(s): May 10, 2019
Received by editor(s) in revised form: July 23, 2019
Published electronically: October 18, 2019
Additional Notes: Part of this research was done while the first author was visiting the Department of Mathematics of Indiana University at Bloomington, supported by the “Salvador de Madariaga” grant no. PRX18/00404 funded by the Spanish Government. He was also partially supported by the research grants MTM2015-63612-P and PGC2018-098321-B-I00, of the Spanish Government.
The second author was supported by NSF grant DMS-1348589.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2019 American Mathematical Society