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Geometrical Conditions for the Existence of a Milnor Vector Field

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Abstract

We introduce several sufficient conditions to guarantee the existence of the Milnor vector field for new classes of singularities of map germs. This special vector field is related with the equivalence problem of the Milnor fibrations for real and complex singularities, if they exit.

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Notes

  1. As have been point out in dos Santos et al. (2020), the existence proof attempt in Cisneros-Molina et al. (2010) for case \( \mathrm{{Disc}}G = \{0\} \) appears to contain a non-removable gap. For more details, see (Ribeiro 2018, Chapter7)

  2. as was made by Milnor in Milnor (1968),

References

  • Cisneros-Molina, J.-L., Seade, J., Snoussi, J.: Refinements of Milnor’s fibration theorem for complex singularities. Adv. Math. 222, 937–970 (2009)

    Article  MathSciNet  Google Scholar 

  • Cisneros-Molina, J.-L., Seade, J., Snoussi, J.: Milnor fibrations and d-regularity for real analytic singularities. Int. J. Math. 21, 419–434 (2010)

    Article  MathSciNet  Google Scholar 

  • dos Santos, R.N.A.: Uniform \((m)\)-condition and strong Milnor fibrations. Singularities II Contemp. Math. Am. Math. Soc. 475, 189–198 (2008)

    MATH  Google Scholar 

  • dos Santos, R.N.A.: Equivalence of real Milnor fibrations for quasi-homogeneous singularities. Rocky Mt. J. Math. 42, 439–449 (2012)

    Article  MathSciNet  Google Scholar 

  • dos Santos, R. N.A., Tibăr, M.: Real map germs and higher open books (2008). arXiv:0801.3328

  • dos Santos, R.N.A., Tibăr, M.: Real map germs and higher open book structures. Geom. Dedic. 147, 177–185 (2010)

    Article  MathSciNet  Google Scholar 

  • dos Santos, R.N.A., Chen, Y., Tibăr, M.: Singular open book structures from real mappings. Cent. Eur. J. Math. 11, 817–828 (2013)

    MathSciNet  MATH  Google Scholar 

  • dos Santos, R.N.A., Ribeiro, M.F., Tibăr, M.: Fibrations of highly singular map germs. Bull. Sci. Math. 155, 92–111 (2019). https://doi.org/10.1016/j.bulsci.2019.05.001

    Article  MathSciNet  MATH  Google Scholar 

  • dos Santos, R.N.A., Ribeiro, M.F., Tibăr, M.: Milnor-Hamm sphere fibrations and the equivalence problem. J. Math. Soc. Jpn. 72(3), 945–957 (2020). https://doi.org/10.2969/jmsj/82278227

    Article  MathSciNet  MATH  Google Scholar 

  • Dutertre, N., dos Santos, R.N.A., Chen, Y., do Espirito Santo, A.: Open book structures on semi-algebraic manifolds. Manuscr. Math. 149(1–2), 205–222 (2016)

  • Dũng Tráng, L.: Some remarks on relative monodromy. Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, pp. 397–403 (1977)

  • Fukuda, T.: Local topological properties of differentiable mappings II. Tokyo J. Math. 8, 501–520 (1985)

    Article  MathSciNet  Google Scholar 

  • Hansen, N. B.: Milnor’s Fibration Theorem for Real Singularities. Master Thesis, Faculty of Mathematics and Natural Sciences University of Oslo (2014)

  • Massey, D. B.: Real analytic Milnor fibrations and a strong Łojasiewicz inequality in Real and complex singularities. Real and complex singularities, London Math. Soc. Lecture Note Ser., vol. 380. Cambridge University Press, Cambridge, pp. 268–292 (2010)

  • Milnor, J.W.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)

    Google Scholar 

  • Oka, M.: Topology of polar weighted homogeneous hypersurface. Kodai Math. J. 31, 163–182 (2008)

    Article  MathSciNet  Google Scholar 

  • Oka, M.: Non degenerate mixed functions. Kodai Math. J. 33, 1–62 (2010)

    Article  MathSciNet  Google Scholar 

  • Oka, M.: On Milnor fibrations of mixed functions, \(a_{f}\)-condition and boundary stability. Kodai Math. J. 38, 581–603 (2015)

    Article  MathSciNet  Google Scholar 

  • Parameswaran, A.J., Tibăr, M.: Thom regularity and Milnor tube fibrations. Bull. Sci. Math. 143, 58–72 (2018). https://doi.org/10.1016/j.bulsci.2017.12.001

    Article  MathSciNet  MATH  Google Scholar 

  • Pichon, A.: Real analytic germs \(f\bar{g}\) and open-book decompositions of the 3-sphere. Int. J. Math. 16, 1–12 (2005)

    Article  Google Scholar 

  • Pichon, A., Seade, J.: Real singularities and open-book decompositions of the 3-sphere. Ann. Fac. Sci. Toulouse Math. 12, 245–265 (2003)

    Article  MathSciNet  Google Scholar 

  • Pichon, A., Seade, J.: Fibred multilinks and singularities \(f\bar{g}\). Math. Ann. 342, 487–514 (2008)

    Article  MathSciNet  Google Scholar 

  • Seade, J.: Open book decompositions associated to holomorphic vector fields. Bol. Soc. Mat. Mexicana 3, 323–336 (1997)

    MathSciNet  MATH  Google Scholar 

  • Ribeiro, M.F.: Singular Milnor Fibrations. PhD thesis, Instituto de Ciências Matemáticas e de Computacão, Universidade de São Paulo, São Carlos (2018). http://www.teses.usp.br/teses/disponiveis/55/55135/tde-06072018-115031/

  • Ruas, M.A.S., Seade, J., Verjovsky, A.: On Real Singularities with a Milnor Fibrations, Trends Math., pp. 191–213. Birkhauser, Basel (2002)

    MATH  Google Scholar 

  • Ruas, M.A.S., dos Santos, R.N.A.: Real Milnor fibrations and (C)-regularity. Manusc. Math. 117, 207–218 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous the Professor Mihai Marius Tibăr for the valuable suggestions. They also thank the anonymous referee for his/her time reading our paper and for all valuable comments, suggestions and corrections. The second author would like to thank the partial support from CNPq grant 313780/2017-0, Fapesp grant 2017/20455-3 and the Fapesp Thematic Project grant 2019/21181-0. Thanks a lot!

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Espirito Santo, Vitória, 29.075-910, Brazil

    Maico F. Ribeiro

  2. Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo, Caixa Postal 668, São Carlos, 13.560-970, Brazil

    Raimundo Nonato Araújo dos Santos

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  1. Maico F. Ribeiro
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  2. Raimundo Nonato Araújo dos Santos
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Correspondence to Maico F. Ribeiro.

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Ribeiro, M.F., Araújo dos Santos, R.N. Geometrical Conditions for the Existence of a Milnor Vector Field. Bull Braz Math Soc, New Series 52, 771–789 (2021). https://doi.org/10.1007/s00574-020-00230-9

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