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The local Euler obstruction and topology of the stabilization of associated determinantal varieties

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Abstract

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney’s \(m_{d}\) multiplicity.

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Acknowledgements

The authors are grateful to Jonathan Mboyo Esole, Thiago de Melo, Otoniel Silva, Jawad Snoussi and Xiping Zhang for their careful reading of the first draft of this paper and for their suggestions. We also thank the referee for his/her careful reading and for his/her suggestions and corrections to the final version of this paper. The first author was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grant PVE 401565/2014-9. The second author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP, Brazil, Grants 2015/16746-7 and 2017/09620-2 and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grants 474289/2013-3, 303641/2013-4 and 303046/2016-3 . The third author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP, Brazil, Grant 2014/00304-2 and Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, Grant 305651/2011-0. This paper was written while the second author was visiting Northeastern University, Boston, USA. During this period the first author had also visited the Universidade de São Paulo at São Carlos, Brazil, and we would like to thank these institutions for their hospitality.

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Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Terence Gaffney

  2. Instituto de Ciências Matemáticas e de Computação-USP, Av. Trabalhador são-carlense, 400-Centro, São Carlos, SP, CEP 13566-590, Brazil

    Nivaldo G. Grulha Jr. & Maria A. S. Ruas

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  1. Terence Gaffney
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  2. Nivaldo G. Grulha Jr.
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  3. Maria A. S. Ruas
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Correspondence to Nivaldo G. Grulha Jr..

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Gaffney, T., Grulha, N.G. & Ruas, M.A.S. The local Euler obstruction and topology of the stabilization of associated determinantal varieties. Math. Z. 291, 905–930 (2019). https://doi.org/10.1007/s00209-018-2141-y

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  • DOI: https://doi.org/10.1007/s00209-018-2141-y

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