Asian Journal of Mathematics

Volume 24 (2020)

Number 5

Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

Pages: 803 – 820



J. Edson Sampaio (Departamento de Matemática, Universidade Federal do Ceará, Pici, Fortaleza, CE, Brazil; and BCAM - Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain )


This paper is devoted to study multiplicity and regularity of complex analytic sets. We present an equivalence for complex analytical sets, named blow-spherical equivalence and we obtain several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski’s multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski’s multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of the complex analytic curves.


regularity, blow-spherical equivalence, complex analytic sets

2010 Mathematics Subject Classification

14B05, 32S05, 32S50

The author was partially supported by CNPq-Brazil grant 303811/2018-8, by the ERCEA 615655 NMST Consolidator Grant and also by the Basque Government through the BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.

Received 22 July 2017

Accepted 29 January 2020

Published 10 March 2021